Write an equation for a linear function given a graph of [latex]y - 0=-3\left(x - 0\right)[/latex] ; [latex]y=-3x[/latex]. Substitute the slope and the coordinates of one of the points into the point-slope form. We are not given the slope of the line, but we can choose any two points on the line to find the slope. The initial value for this function is 200 because he currently owns 200 songs, so \(N(0)=200\), which means that \(b=200\). The slope-intercept equation of the line is \(y=2x9\). The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. Write a formula for the number of songs, The pressure as a function of depth equals four hundred thirty-four thousandths times depth plus fourteen and six hundred ninety-six thousandths. Lets figure out what we know from the given information. x do to the right-hand side-- and we are in standard form. \[\begin{align*} y-y_1&=m(x-x_1) \\ y-1&=\dfrac{1}{3}(x-0) \end{align*}\]. Then we use algebra to find the slope-intercept form. , +3 Determine the units for output and input values. For the following exercises, determine whether each function is increasing or decreasing. x and 1 www.chinahighlights.com/shangglev-train.htm Consider, for example, the first commercial maglev train in the world, the Shanghai MagLev Train (Figure 1). And now to get it in slope If we do that, what do we get? f(5)=1 We can convert it to the slope-intercept form as shown. Each year in the decade of the 1990s, average annual income increased by $1,054. Up until now, we have been using the slope-intercept form of a linear equation to describe linear functions. to use point slope form. subscript 1 and a y with a subscript 1, that's like saying x We know that [latex]m=2[/latex]and that [latex]{x}_{1}=4[/latex]and [latex]{y}_{1}=1[/latex]. The point-slope equation of the line is \(y_21=2(x_25)\). algebraically manipulate this guy right here to put it into Then we use algebra to find the slope-intercept form. and (4,10), Passes through are licensed under a, Introduction to Polynomial and Rational Functions, Introduction to Exponential and Logarithmic Functions, Graphs of the Other Trigonometric Functions, Introduction to Trigonometric Identities and Equations, Solving Trigonometric Equations with Identities, Double-Angle, Half-Angle, and Reduction Formulas, Sum-to-Product and Product-to-Sum Formulas, Introduction to Further Applications of Trigonometry, Introduction to Systems of Equations and Inequalities, Systems of Linear Equations: Two Variables, Systems of Linear Equations: Three Variables, Systems of Nonlinear Equations and Inequalities: Two Variables, Solving Systems with Gaussian Elimination, Sequences, Probability and Counting Theory, Introduction to Sequences, Probability and Counting Theory, Finding Limits: Numerical and Graphical Approaches, Shanghai MagLev Train (credit: kanegen/Flickr), The slope of a function is calculated by the change in, http://www.chinahighlights.com/shanghai/transportation/maglev-train.htm, https://openstax.org/books/precalculus-2e/pages/1-introduction-to-functions, https://openstax.org/books/precalculus-2e/pages/2-1-linear-functions, Creative Commons Attribution 4.0 International License. Want to cite, share, or modify this book? Now we will learn another way to write a linear function called point-slope form which is given below: yy1 = m(xx1) y y 1 = m ( x x 1) where m m is the slope of the linear function and (x1,y1) ( x 1, y 1) is any point which satisfies the linear function. Identify two points on the line, such as ( \\ y&=\dfrac{1}{3}x+1 &\text{Add 1 to each side.} The slope, or rate of change, of a function mm can be calculated according to the following: where Because we are told that the population increased, we would expect the slope to be positive. Write the point-slope form of an equation of a line that passes through the points (5, 1) and (8, 7). To restate the function in words, we need to describe each part of the equation. Plug them into the slope-intercept form for a linear function. [latex]\begin{array}{llll}y - 4=-\frac{1}{2}\left(x - 6\right)\hfill & \hfill \\ y - 4=-\frac{1}{2}x+3\hfill & \text{Distribute the }-\frac{1}{2}.\hfill \\ \text{}y=-\frac{1}{2}x+7\hfill & \text{Add 4 to each side}.\hfill \end{array}[/latex]. x Wed love your input. d, If you are redistributing all or part of this book in a print format, A clown at a birthday party can blow up . 4 we can calculate the slope Write the point-slope form of an equation of a line that passes through the points[latex]\left(5,\text{ }1\right)[/latex] and [latex]\left(8,\text{ }7\right)[/latex]. Last week he sold 3 new policies, and earned $760 for the week. y It carries passengers comfortably for a 30-kilometer trip from the airport to the subway station in only eight minutes. We know that \(m=2\) and that \(x_1=4\) and \(y_1=1\). C( The initial value, 14.696, is the pressure in PSI on the diver at a depth of 0 feet, which is the surface of the water. to negative 2/3 x. Lets use (0, 1) for our point. [latex]\begin{array}{l}y - 1=2\left(x - 5\right)\hfill \\ y - 1=2x - 10\hfill \\ \text{}y=2x - 9\hfill \end{array}[/latex]. Studies from the early 2010s indicated that teens sent about 60 texts a day, while more recent data indicates much higher messaging rates among all users, particularly considering the various apps with which people can communicate. and the linear equation would be The number of songs increases by 15 songs per month, so the rate of change is 15 songs per month. To graph, you must plug in 0 for either x or y to get the y- or x-intercept. ( to simplify this? x ). Both equations describe the line shown in Figure \(\PageIndex{8}\). If we did not notice the rate of change from the table we could still solve for the slope using any two points from the table. When using standard form, a a, b b, and c c are all replaced with real numbers. If we discover two points we only have to devise them on the graph merge them by a line and spread from both flanks. Twelve minutes after leaving, she is 0.9 miles from home. Direct link to CubanMissileCrisis's post At 7:25,Sal says that the, Posted 10 years ago. Now, we can literally just For the viewing window, set the minimum value of xx to be 1010 and the maximum value of xx to be 10.10. a,b+1 Then you can solve it like a regular equation and you would get y =-12. The function is increasing because [latex]m>0[/latex]. Keep in mind that the slope-intercept form and the point-slope form can be used to describe the same function. how would you know if the line is a parrallel line, well if slope of line 1 is equal to slope of line 2 they are parallel, I'm not sure, but the way I learned it, you are right. Use the table to write a linear equation. y Interpret the slope as the change in output values per unit of the input value. Up until now, we have been using the slope-intercept form of a linear equation to describe linear functions. ) If we view this as our end ). Slope intercept form is y is This positive slope we calculated is therefore reasonable. The slope determines if the function is an increasing linear function, a decreasing linear function, or a constant function. The function is increasing because Interpret slope as a rate of change. The rate of change relates the change in population to the change in time. Write an equation, P(t),P(t), for the population tt years after 2003. and Another approach to representing linear functions is by using function notation. f(5)=2 An example of slope could be miles per hour or dollars per day. We can write the formula N(t)=15t+200.N(t)=15t+200. Site: http://mathispower4u.com We could also write the slope as [latex]m=0.6[/latex]. on any value. 1 This gives us the linear function y = 2 3 x + 1 In many cases the value of b is not as easily read. Negative 2 plus 6 is plus 4. the denominator by 3. Notice the units appear as a ratio of units for the output per units for the input. A teen has an unlimited number of texts in his or her data plan for a cost of $50 per month. But point slope form says Keeping track of units can help us interpret this quantity. Use the two points to calculate the slope. 2 ). Lets begin by describing the linear function in words. Now we can use the point to find the y-intercept by substituting the given values into the slope-intercept form of a line and solving for b. b. g (x) . add 6 to both sides of this equation. \\ y&=2x7 &\text{Add 1 to each side.} \(m=\frac{1,8681,442}{2,0122,009} = \frac{426}{3} =\text{ 142 people per year}\). (0,b). algebraically manipulate it so that the x's and the We can interpret this as Ilyas base salary for the week, which does not depend upon the number of policies sold. If f(x)f(x) is a linear function, with An increasing linear function results in a graph that slants upward from left to right and has a positive slope. \[\begin{align*} m&=\dfrac{y_2-y_1}{x_2-x_1} \\ &=\dfrac{2-1}{3-0} \\ &=\dfrac{1}{3} \end{align*}\]. in orange. If Ben produces 100 items in a month, his monthly cost is represented by, \[\begin{align*} C(100)&=1250+37.5(100) \\ &=5000 \end{align*}\]. ( A boat is 100 miles away from the marina, sailing directly toward it at 10 miles per hour. One type of function notation is the slope-intercept form of an equation. See Figure \(\PageIndex{7}\). value-- that is that 6 right there, or that 6 right there-- Here's a link to a screenshot of an example: http://imgur.com/UC1j1su 1 To find the slope use the formula m = (y2 - y1) / (x2 - x1) where (x1, y1) and (x2, y2) are two points on the line. +1. When x changed by 4, y changed by negative 1. In the example of the train, we might use the notation The point-slope form of a linear equation takes the form. and So if you give me one of them, We can see from the graph in Figure 3 that the y-intercept in the train example we just saw is Or when y changed by negative 1, x changed by 4. N, Write a linear function CC where C(x)C(x) is the cost for xx items produced in a given month. (1,4) )=2 The rate of change relates the change in population to the change in time. The rate, \(m\), is 83 meters per second. A phone company charges for service according to the formula: C(n)=24+0.1n,C(n)=24+0.1n, where nn is the number of minutes talked, and C(n)C(n) I'll take this one-- so y minus the y value over here, so y The rate of change, which is constant, determines the slant, or slope of the line. [latex]\begin{array}{llll} y - 1=2\left(x - 4\right)\hfill & \hfill \\ y - 1=2x - 8\hfill & \text{Distribute the }2.\hfill \\ \text{}y=2x - 7\hfill & \text{Add 1 to each side}.\hfill \end{array}[/latex]. y 2 y1=f(x1)y1=f(x1) and y2=f(x2),y2=f(x2), so we could equivalently write. (5,2), (8,2) 2 We will describe the trains motion as a function using each method. The point-slope form is derived from the slope formula. (4,6) \[\begin{align*} y-1&=2(x-5) \\ y-1&=2x-10 \\ y&=2x-9 \end{align*}\]. \[\begin{align*} y1&=2(x4) \\ y1&=2x8 &\text{Distribute the 2.} Remember, a y-intercept will always have an X-value = 0 because the point must sit on the y-axis. The rate of change for this example is constant, which means that it is the same for each input value. If ff is a linear function, f(0.1)=11.5,andf(0.4)=5.9,f(0.1)=11.5,andf(0.4)=5.9, find an equation for the function. Is this function increasing or decreasing? (1,3) Customer Voice Questionnaire FAQ (4,10) where [latex]\Delta y[/latex] is the change in output and [latex]\Delta x[/latex] is the change in input. However, we often need to calculate the slope given input and output values. In the Linear and nonlinear functions exercise, there is a type of question which displays an equation not in linear format and asks if the given equation can be expressed as a linear equation. For the train problem we just considered, the following word sentence may be used to describe the function relationship. The costs that can vary include the cost to produce each item, which is $37.50 for Ben. We are not given the slope of the line, but we can choose any two points on the line to find the slope. So the function is ( If ( I think y=mx+b is the easiest formula. \(m=\frac{43}{02} =\frac{1}{-2}=-\frac{1}{2}\); decreasing because \(m<0\). m Example 1. The input consists of non-negative real numbers. So the function is \(f(x)=\frac{3}{4}x+7\), and the linear equation would be \(y=\frac{3}{4}x+7\). is figure out the slope. x6 Identify two points on the line, such as \((0, 2)\) and \((2,4)\). Another approach to representing linear functions is by using function notation. The train began moving at this constant speed at a distance of 250 meters from the station. unitsfortheinput This makes sense because we can see from Figure 11 that the line crosses the y-axis at the point ( 0,2)( 0,2), which is the y-intercept, so b=2.b=2. which is 0, y ends up at the 0, and y was at 6. Terry's elevation, E(t),E(t), in feet after tt seconds is given by E(t)=300070t.E(t)=300070t. 0,1 x=0 and Accessibility StatementFor more information contact us atinfo@libretexts.org. x When temperature is 0 degrees Celsius, the Fahrenheit temperature is 32. Maybe it's time for you to find some of your own "inner nerd". The greater the absolute value of the slope, the steeper the line is. 6,1 Here's the graph we're given: We can see that the slope of . Write the point-slope form of an equation of a line that passes through the points \((5,1)\) and \((8, 7)\). so that the . We can substitute these values into the general point-slope equation. Rather than solving for m,m, we can tell from looking at the table that the population increases by 80 for every 2 weeks that pass. We also know one point, so we know \(x_1=6\) and \(y_1 =1\). Calculate and interpret slope. Interpret the slope as the change in output values per unit of the input value. There are several ways to represent a linear function, including word form, function notation, tabular form, and graphical form. One example of function notation is an equation written in the form known as the slope-intercept form of a line, where 1999-2023, Rice University. Well, we have our end point, That information may be provided in the form of a graph, a point and a slope, two points, and so on. \\ y&=3x19 &\text{Simplify to slope-intercept form.} The domain is comprised of all real numbers because any number may be doubled, and then have one added to the product. Determine whether a linear function is increasing, decreasing, or constant. You get a y is equal Therefore, \(b=7\). The point-slope form of an equation is also useful if we know any two points through which a line passes. 2/3 times x minus our x-coordinate. So, our finishing y point is 0, . should have-- the left-hand side of this equation is what? These are the same equations, ( C( This tells us that the pressure on the diver increases 0.434 PSI for each foot her depth increases. y As an Amazon Associate we earn from qualifying purchases. ) The point-slope form is useful for finding a linear equation when given the slope of a line and one point. In 1989 the population was 275,900. Given a linear function, graph by plotting points. (0,0). is the initial or starting value of the function (when input, we do the example. a. Graph the linear function ff where f(x)=ax+bf(x)=ax+b on the same set of axes on a domain of [ 4,4 ][ 4,4 ] for the following values of aa and b.b. Let's do that. ( e.g. The coordinate pairs are We can then use the points to calculate the slope. 6,1 Now we can substitute these values into the general point-slope equation. You divide the numerator and Lets choose In the examples we have seen so far, we have had the slope provided for us. Write a complete sentence describing Marias starting elevation and how it is changing over time. we can manipulate it to get any of the other ones. ) that in this video, if the point negative 3 comma 6 is on For example, suppose we are told that a line has a slope of 2 and passes through the point [latex]\left(4,1\right)[/latex]. and represents the distance of the train from the station when it began moving at a constant speed. Graph linear functions. So let's do slope intercept For a decreasing function, the slope is negative. The point-slope form is derived from the slope formula. ) Graphing of a linear function As we comprehend to plot a line on a graph we require any two points on it. [latex]\frac{4,400\text{ people}}{4\text{ years}}=1,100\text{ }\frac{\text{people}}{\text{year}}[/latex]. The point-slope form is particularly useful if we know one point and the slope of a line. Direct link to StudyBuddy's post Just what Kim said. We can use the function relationship from above, a function with a positive slope: If \(f(x)=mx+b\), then \(m>0\). A line with a slope of zero is horizontal as in Figure 5(c). \[\begin{align*} y-1&=\dfrac{1}{3}(x-0) \\ y-1&=\dfrac{1}{3}x &\text{Distribute the }\dfrac{1}{3}. 2/3 x times 3 is just 2x. 0,2) and (2,4).(2,4). I've had that happen as well when I click on someone's post. To find the rate of change, we divide the change in output by the change in input. The letter x x represents the independent variable and the letter y y represents the dependent variable. However, we often need to calculate the slope given input and output values. \[ \begin{align*} &m=\dfrac{y-y_1}{x-x_1} &\text{assuming }x{\neq}x_1 \\ &m(x-x_1)=\dfrac{y-y_1}{x-x_1}(x-x_1) &\text{Multiply both sides by }(x-x_1). 3 Because we are told that the population increased, we would expect the slope to be positive. Legal. and 1 So this 0, we have that 0, that Write the equation of a line parallel or perpendicular to a given line. a particular value x and a particular value of y, Our finishing x-coordinate 0,4 The population of a small town increased from 1,442 to 1,868 between 2009 and 2012. y As noted earlier, the order in which we write the points does not matter when we compute the slope of the line as long as the first output value, or y-coordinate, used corresponds with the first input value, or x-coordinate, used. In function notation, [latex]{y}_{1}=f\left({x}_{1}\right)[/latex] and [latex]{y}_{2}=f\left({x}_{2}\right)[/latex] so we could write: [latex]m=\frac{f\left({x}_{2}\right)-f\left({x}_{1}\right)}{{x}_{2}-{x}_{1}}[/latex]. A third method of representing a linear function is through the use of a table. We can use these points to calculate the slope. For an increasing function, as with the train example, the output values increase as the input values increase. You would plug in 0 for x. Determine the units for output and input values. Use the model to make a prediction by evaluating the function at a given x-value. [latex]\begin{array}{l}y-{y}_{1}=m\left(x-{x}_{1}\right)\\ y - 1=2\left(x - 5\right)\end{array}[/latex]. 2,2 these, these are just three different ways of writing Determine whether lines are parallel or perpendicular given their equations. our starting y point is 6. We can see from the table that the initial value for the number of rats is 1000, so \(b=1000\). ( m=2. 2 x+7. Then rewrite it in the slope-intercept form. We go through two different examples for writing the equatio. A new plant food was introduced to a young tree to test its effect on the height of the tree. ) So the equation would be 8*0 -2y =24, or -2y =24. =1. Q/A? Is the initial value always provided in a table of values like Table \(\PageIndex{1}\)? ( Here, we will learn another way to write a linear function, the point-slope form. For example, suppose we are given an equation in point-slope form, negative 2/3 x. ( y look extra clean and have no fractions here, we could in his collection as a function of time, t,t, the number of months. This is more of a question for Kim or someone who knows about Khan Academy: Just what Kim said. Then rewrite the equation in the slope-intercept form. ( Jessica is walking home from a friends house. Marcus currently has 200 songs in his music collection. Note in function notation two corresponding values for the output y1 y1 and y2y2 for the function Note in function notation two corresponding values for the output \(y_1\) and \(y_2\) for the function \(f\), \(y_1=f(x_1)\) and \(y_2=f(x_2)\), so we could equivalently write, \[m=\dfrac{f(x_2)-f(x_1)}{x_2-x_1} \nonumber\]. Timmy goes to the fair with $40. plus 2/3 x to both sides of this equation. Recognize the standard form of a linear function. [latex]\begin{array}{llll}y+1=3\left(x - 6\right)\hfill & \hfill \\ y+1=3x - 18\hfill & \text{Distribute 3}.\hfill \\ \text{}y=3x - 19\hfill & \text{Simplify to slope-intercept form}.\hfill \end{array}[/latex]. Find an equation for \(I(n)\), and interpret the meaning of the components of the equation. A clothing business finds there is a linear relationship between the number of shirts, n,n, it can sell and the price, p,p, it can charge per shirt. f, [ 10,10 ]:f(x)=0.02x0.01. The slope, or rate of change, of a function \(m\) can be calculated according to the following: \[m=\dfrac{\text{change in output (rise)}}{\text{change in input (run)}}=\dfrac{{\Delta}y}{{\Delta}x}=\dfrac{y_2-y_1}{x_2-x_1}\]. 4 Given a graph of a line, we can write a linear function in the form y=mx+b by identifying the slope (m) and y-intercept (b) in the graph. For a decreasing function, the slope is negative. x ), and A line with a positive slope slants upward from left to right as in Figure \(\PageIndex{5}\)(a). x+7, 2,2 Determine whether a linear function is increasing, decreasing, or constant. Standard Form (Linear Equation): ax+by=c ax +by = c. The letters a a, b b, and c c are all coefficients. Maria's elevation, E(t),E(t), in feet after tt minutes is given by OpenStax is part of Rice University, which is a 501(c)(3) nonprofit. In this case, the slope is negative so the function is decreasing. Example: Matching Linear Functions to Their Graphs The coordinate pairs are [latex]\left(3,-2\right)[/latex]and [latex]\left(8,1\right)[/latex]. b ( Suppose, for example, we know that a line passes through the points \((0, 1)\) and \((3, 2)\). Is this function increasing or decreasing? m>0. The initial value, 14.696, is the pressure in PSI on the diver at a depth of 0 feet, which is the surface of the water. For the following exercises, given each set of information, find a linear equation satisfying the conditions, if possible. I'm just saying, if we go from was our starting y point, that is that 6 right there. So the left-hand side of the m, Lets figure out what we know from the given information. standard form, this is the standard form of the equation. A linear equation can be written in slope-intercept form, y = m x + b, where the y and x values of the equation are related by the slope, or steepness, of the line, m, and the point at which. 3, and we're done. Both equations describe the line shown in Figure 8. ( m, . \[\begin{align*} y-4&=-\dfrac{1}{2}(x-6) \\ y-4&=-\dfrac{1}{2}x+3 &\text{Distribute the }-\dfrac{1}{2}. x 0,2)( x , Consider the graph of the line \(f(x)=2x+1\). You also already have an ordered pair disguised as part of the given function. Graph the linear function ff on a domain of [ 10,10 ][ 10,10 ] for the function whose slope is 1818 and y-intercept is 31163116. Write the domain in interval form, if possible. This makes sense because the number of texts remaining decreases each day and this function represents the number of texts remaining in the data plan after \(x\) days. If \(f(x)\) is a linear function, and \((3,2)\) and \((8,1)\) are points on the line, find the slope. So in the equation that I said, let's find the y-intercept first. Given two values for the input, [latex]{x}_{1}[/latex] and [latex]{x}_{2}[/latex], and two corresponding values for the output, [latex]{y}_{1}[/latex]and [latex]{y}_{2}[/latex] which can be represented by a set of points, [latex]\left({x}_{1}\text{, }{y}_{1}\right)[/latex]and [latex]\left({x}_{2}\text{, }{y}_{2}\right)[/latex], we can calculate the slope [latex]m[/latex],as follows: [latex]m=\frac{\text{change in output (rise)}}{\text{change in input (run)}}=\frac{\Delta y}{\Delta x}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}[/latex]. Use the points to calculate the slope. ( So the equation would be 8*0 -2y =24, or -2y =24. unitsfortheoutput So, for example, and we'll do Write and interpret an equation for a linear function. y= Label the points for the input values of 1010 and 10.10. x Lets begin by describing the linear function in words. Recall that the slope measures steepness. is that 0 right there. Left-hand side of the equation, Find the slope. ( To write the rule of a function from the table is somehow tricky but can be made easier by . If We can simplify it We have a point, we could pick Both equations describe the line graphed below. 5). If we wanted to rewrite the equation in slope-intercept form, we apply algebraic techniques. Then rewrite it in the slope-intercept form. There are several ways to represent a linear function, including word form, function notation, tabular form, and graphical form. in which the total distance \[\begin{align*} y-y_1&=m(x-x_1) \\ y-(-4)&=3(x-(-2)) \\ y+4 &= 3(x+2)\end{align*}\]. We can use the coordinates of the two points to find the slope. Graph linear functions. How can we analyze the trains distance from the station as a function of time? Then, determine whether the graph of the function is increasing, decreasing, or constant. We start by finding the rate of change. Despite the fact that you just called me a "nerd" in a comment you posted on another entry, I'm going to try and answer you question anyway. Keeping track of units can help us interpret this quantity. So \(m=2\). The population increased by 27,80023,400=4,400 1 x For an increasing function, as with the train example, the output values increase as the input values increase. m sides of this equation. which can be represented by a set of points, Given the graph of a linear function, write an equation to represent the function. How many songs will he own in a year? x Therefore we know that \(m=15\). Use the two points to calculate the slope. Write the point-slope form of an equation of a line that passes through the points As before, we can use algebra to rewrite the equation in the slope-intercept form. The input represents time, so while nonnegative rational and irrational numbers are possible, negative real numbers are not possible for this example. Now we can substitute these values into point-slope form. \[\begin{align*} m&=\dfrac{y_2-y_1}{x_2-x_1} \\ &=\dfrac{4-7}{4-0} \\&=-\dfrac{3}{4}\end{align*}\]. Posted 11 years ago. is a linear function, and Direct link to Abhik Pal's post Not necessarily. The slope is 3, so m= 3. y=2x9. If we wanted to find the slope-intercept form without first writing the point-slope form, we could have recognized that the line crosses the y-axis when the output value is 7.
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