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1 Chapter 1. Write a Function in Slope-Intercept Form CHAPTER 1 Write a Function in Slope-Intercept Form Here you ll learn how to write the slope-intercept form of linear functions and how to work with functions in this form. What if the linear function W(g) represented a family s monthly water bill, with g as the number of gallons of water used. If you knew what the function was, could you find W(25)? How about if you knew the slope of the function and the value of W(25)? Could you determine what the function was? Suppose you knew the values of W(25) and W(50). Could you determine the function in this case? After completing this Concept, you ll be able to perform tasks like these. Guidance Remember that a linear function has the form f (x)=mx + b. Here f (x) represents the y values of the equation or the graph. So y = f (x) and they are often used interchangeably. Using the functional notation in an equation often provides you with more information. For instance, the expression f (x)= mx + b shows clearly that x is the independent variable because you substitute values of x into the function and perform a series of operations on the value of x in order to calculate the values of the dependent variable, y. In this case when you substitute x into the function, the function tells you to multiply it by m and then add b to the result. This process generates all the values of y you need. Example A Consider the function f (x)=3x 4.Find f (2), f (0),and f ( 1). Each number in parentheses is a value of x that you need to substitute into the equation of the function. f (2)=2; f (0)= 4; and f ( 1)= 7 Function notation tells you much more than the value of the independent variable. It also indicates a point on the graph. For example, in the above example, f ( 1) = 7. This means the ordered pair ( 1, 7) is a solution to f (x)=3x 4 and appears on the graphed line. You can use this information to write an equation for a function. 1

2 Example B Write an equation for a line with m = 3.5 and f ( 2)=1. You know the slope, and you know a point on the graph, ( 2, 1). Using the methods presented in this Concept, write the equation for the line. Begin with slope-intercept form. y = mx + b Substitute the value for the slope. y = 3.5x + b Use the ordered pair to solve for b. 1 = 3.5( 2)+b b = 8 Rewrite the equation. y = 3.5x + 8 or f (x)=3.5x + 8 Example C Write an equation for a line with f ( 1)=2 and f (5)=20. You know two points on the graph. Using the methods presented in the previous Concept, write the equation for the line. First, you must find the slope: m = y 2 y 1 = ( 1) = 18 6 = 3. Now use the slope-intercept form: y = mx + b Substitute the value for the slope. y = 3x + b Use the ordered pair to solve for b. 2 = 3( 1)+b b = 5 Rewrite the equation. y = 3x + 5 or f (x)=3x + 5 Vocabulary Slope: The slope of a line is the vertical change, f (x), divided by the horizontal change, x. The slope of a line measures its steepness (either negative or positive). The formula for slope is: slope = f (x) x = rise run = f (x 2) f (x 1 ) where (x 1,y 1 ) and (x 2,y 2 ) are any two points on the line. Slope-intercept form: The slope-intercept form of a function is: f (x)=(slope)x +(y intercept) or f (x)=(m)x + b, where m = slope and b = y intercept. Zero slope: A line with zero slope is a line without any steepness, or a horizontal line. Undefined slope: An undefined slope cannot be computed. Vertical lines have undefined slopes. 2

3 Chapter 1. Write a Function in Slope-Intercept Form Guided Practice Write an equation for a line with f (0)=2 and f (3)= 4 and use it to find f ( 5), f (2), f (0), and f (z). Notice that the first point given as an input value is 0, and the output is 2, which means the point is (0,2). This is the y-intercept. So, all we have to do is find the slope and then plug both values into the slope-intercept form: m = y 2 y 1 = = 6 3 = 2. Now use the slope-intercept form. y = mx + b Substitute the value for the slope. y = 2x + b Substitute the value for the y-intercept y = 2x + 2 or f (x)= 2x + 2 Now we find the values of f ( 5), f (2), f (0), and f (z) for f (x)= 2x + 2. f ( 5)= 2( 5)+2 = = 12 f (2)= 2(2)+2 = 2 f (0)= 2(0)+2 = 0 f (z)= 2z + 2 Practice Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Linear Equations inslope-interceptform (14:58) MEDIA Click image to the left for more content. 1. Consider the function f (x)= 2x 3.Find f ( 3), f (0),and f (5). 2. Consider the function f (x)= 2 3x + 10.Find f ( 9), f (0),and f (9). In 3 10, find the equation of the linear function in slope intercept form. 3. m = 5, f (0)= 3 4. m = 2, f (0)=5 5. m = 7, f (2)= 1 6. m = 1 3, f ( 1)= m = 4.2, f ( 3)= f 1 4 = 3 4, f (0)= 5 4 3

4 9. f (1.5)= 3, f ( 1)=2 10. f ( 1)=1, f (1)= 1 Mixed Review 11. Translate into a sentence: 4( j + 2) = Evaluate The formula to convert Fahrenheit to Celsius is C(F)= F What is the Celsius equivalent to 35 F? 14. Find the rate of change: The diver dove 120 meters in 3 minutes. 15. What percent of 87.4 is 106? 16. Find the percent of change: The original price was $ The new price is $ Solve for w : 606 = 0.045(w 4000)+0.07w. 4

5 Chapter 2. Write an Equation Given the Slope and a Point CHAPTER 2 Write an Equation Given the Slope and a Point Here you ll be given the slope of a line and a point on the line and you ll learn to write the equation of the line. Suppose that you sent out a text message to all of your friends, asking them what information was needed to write the equation of a line. One of your friends responded that all you need is the slope of the line and a point on the line. Do you think that your friend was correct? If so, does it matter what point you have, and how could you use this information to come up with the equation? In this Concept, you ll get answers to these questions so that you can judge the merits of your friend s advice. Guidance Previously, you learned how to graph solutions to two-variable equations in slope-intercept form. This Concept focuses on how to write an equation for a graphed line when given the slope and a point. There are two things you will need from the graph to write the equation in slope-intercept form: 1. The y intercept of the graph 2. The slope of the line Having these two pieces of information will allow you to make the appropriate substitutions in the slope-intercept formula. Recall the following: Slope-intercept form: y =(slope)x +(y intercept) or y = mx + b Example A Write the equation for a line with a slope of 4 and a y intercept of (0, 3). Slope-intercept form requires two things: the slope and y intercept. To write the equation, you substitute the values into the formula. y =(slope)x +(y intercept) y = 4x +( 3) y = 4x 3 You can also use a graphed line to determine the slope and y intercept. Example B Use the graph below to write an equation in slope-intercept form. 5

www.ck12.org The y intercept is (0, 2). Using the slope triangle, you can determine the slope is rise value 2 for b and the value 3 for m, the equation for this line is y = 3x + 2.

6 The y intercept is (0, 2). Using the slope triangle, you can determine the slope is rise value 2 for b and the value 3 for m, the equation for this line is y = 3x + 2. Writing an Equation Given the Slope and a Point run = 3 1 = 3 1. Substituting the You will not always be given the y intercept, but sometimes you will be given any point on the line. When asked to write the equation given a graph, it may be difficult to determine the y intercept. Perhaps the y intercept is rational instead of an integer. Maybe all you have is the slope and an ordered pair. You can use this information to write the equation in slope-intercept form. To do so, you will need to follow several steps. Step 1: Begin by writing the formula for slope-intercept form: y = mx + b. Step 2: Substitute the given slope for m. Step 3: Use the ordered pair you are given (x,y) and substitute these values for the variables x and y in the equation. Step 4: Solve for b (the y intercept of the graph). Step 5: Rewrite the original equation in Step 1, substituting the slope for m and the y intercept for b. Example C Write an equation for a line with a slope of 4 that contains the ordered pair ( 1, 5). Step 1: Begin by writing the formula for slope-intercept form. Step 2: Substitute the given slope for m. y = mx + b y = 4x + b Step 3: Use the ordered pair you are given, ( 1, 5), and substitute these values for the variables x and y in the equation. 6

7 Chapter 2. Write an Equation Given the Slope and a Point Step 4: Solve for b (the y intercept of the graph). 5 =(4)( 1)+b 5 = 4 + b = b 9 = b Step 5: Rewrite y = mx + b, substituting the slope for m and the y intercept for b. y = 4x + 9 Vocabulary Slope: The slope of a line is the vertical change, y, divided by the horizontal change, x. The slope of a line measures its steepness (either negative or positive). The formula for slope is: slope = y x = rise run Slope-intercept form: The slope-intercept form of an equation is: y =(slope)x +(y intercept) or y =(m)x + b, where m = slope and b = y intercept. Zero slope: A line with zero slope is a line without any steepness, or a horizontal line. Undefined slope: An undefined slope cannot be computed. Vertical lines have undefined slopes. Guided Practice Write the equation for a line with a slope of 3 containing the point (3, 5). Using the five-steps from above: y =(slope)x +(y intercept) y = 3x + b 5 = 3(3)+b 5 = 9 + b 4 = b y = 3x + 4 Practice Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Linear Equations inslope-interceptform (14:58) 7

8 MEDIA Click image to the left for more content. 1. What is the formula for slope-intercept form? What do the variables m and b represent? 2. What are the five steps needed to determine the equation of a line given the slope and a point on the graph (not the y intercept)? In 3 13, find the equation of the line in slope intercept form. 3. The line has a slope of 7 and a y intercept of The line has a slope of 5 and a y intercept of The line has a slope of -2 and a y intercept of The line has a slope of 2 3 and a y intercept of The line has a slope of 1 4 and contains the point (4, 1). 8. The line has a slope of 2 3 and contains the point 1 2,1. 9. The line has a slope of 1 and contains the point 4 5, The slope of the line is 2 3, and the line contains the point (2, 2). 11. The slope of the line is 3, and the line contains the point (3, 5)

9 Chapter 2. Write an Equation Given the Slope and a Point 13. 9

10 CHAPTER 3 Write an Equation Given Two Points Here you ll be given two points and you ll learn how to write the equation of the line that passes through them. Suppose two travelers were lost in a forest. From the same spot, one person traveled 5 miles east and 10 miles south, while the other person traveled 2 miles west and 8 miles north. If a coordinate plane were transposed on top of the forest, with the line going from west to east as the x-axis and the line going from north to south as the y-axis, could you write the equation of the line that passes through the points representing the travelers new locations? After completing this Concept, you ll be able to solve these types of problems. Guidance In many cases, especially real-world situations, you are given neither the slope nor the y intercept. You might have only two points to use to determine the equation of the line. To find an equation for a line between two points, you need two things: 1. The y intercept of the graph 2. The slope of the line Previously, you learned how to determine the slope between two points. Let s repeat the formula here. The slope between any two points (x 1,y 1 ) and (x 2,y 2 ) is: slope = y 2 y 1. The procedure for determining a line given two points is the same five-step process as writing an equation given the slope and a point. Example A Write the equation for the line containing the points (3, 2) and ( 2, 4). You need the slope of the line. Find the line s slope by using the formula. Choose one ordered pair to represent (x 1,y 1 ) and the other ordered pair to represent (x 2,y 2 ). slope = y 2 y 1 x 2 x 1 = = 2 5 Now use the five-step process to find the equation for this line. Step 1: Begin by writing the formula for slope-intercept form. y = mx + b Step 2: Substitute the given slope for m. y = 2 5 x + b 10

11 Chapter 3. Write an Equation Given Two Points Step 3: Use one of the ordered pairs you are given, ( 2, 4), and substitute these values for the variables x and y in the equation. 4 = Step 4: Solve for b (the y intercept of the graph). 2 ( 2)+b 5 4 = b = b 16 5 = b Step 5: Rewrite y = mx + b, substituting the slope for m and the y intercept for b. y = 2 5 x Example B Write the equation for a line containing the points ( 4, 1) and ( 2, 3). 1. Start with the slope intercept form of the line, y = mx + b. 2. Find the slope of the line: m = y 2 y 1 = ( 4) = 2 2 = Substitute the value of the slope for m : y =(1)x + b. 4. Substitute the coordinates ( 2, 3) into the equation for the variables x and y :3= 2 + b b = Rewrite the equation, substituting the slope for m and the y intercept for b: y = x + 5. Example C Write the equation of the line containing the points (3,6) and (-2, 6). 1. Start with the slope intercept form of the line y = mx + b. 2. Find the slope of the line: m = y 2 y 1 = (3) = 0 5 = Substitute the value of the slope in for m : y =(0)x + b y = b. Notice that this is an equation where y equals some number. This means it is a horizontal line. This makes sense since the slope is zero and a horizontal line has a slope of zero. 1. Substitute the coordinates (3, 6) into the equation for the variables x and y :6=(0)3 + b b = Rewrite the equation, substituting the slope for m and the y intercept for b: y = 6. We can see when this will happen in the future, without having to do all the work. Because the two y values were the same, this must be a horizontal line. 11

12 Vocabulary Slope: The slope of a line is the vertical change, y, divided by the horizontal change, x. The slope of a line measures its steepness (either negative or positive). The formula for slope is: slope = y x = rise run = y 2 y 1 where (x 1,y 1 ) and (x 2,y 2 ) are any two points on the line. Slope-intercept form: The slope-intercept form of an equation is: y =(slope)x +(y intercept) or y =(m)x + b, where m = slope and b = y intercept. Zero slope: A line with zero slope is a line without any steepness, or a horizontal line. Undefined slope: An undefined slope cannot be computed. Vertical lines have undefined slopes. Guided Practice Write the equation of the line containing the points (2, -3) and (2, 10). In this case, notice that the two x values are the same. What does this mean? Remember, that for a vertical line, the x value stays the same no matter what the y value is. Since we are trying to write an equation of a line, and in both cases x = 2, we can conclude that our equation is: x = 2 Note: If we were to calculate the slope of the line given the two points, we would get the following: m = y 2 y 1 = 3 ( 2) 2 (2) = 1 0 = undefined. Since our slope is undefined, it must be a vertical line. We would come to the same conclusion that this is a vertical line where x = 2. Practice Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set. However, the practice exercise is the same in both. CK-12 Ba sic Algebra:Linear Equations inslope-interceptform (14:58) MEDIA Click image to the left for more content. 1. What is the first step in finding the equation of a line given two points? In 2 7, find the equation of the line in slope intercept form The line containing the points (2, 6) and (5, 0). 3. The line containing the points (5, 2) and (8, 4).

www.ck12.org Chapter 3. Write an Equation Given Two Points 4. The line containing the points (3, 5) and ( 3, 0). 5. The line containing the points (10, 15) and (12, 20). 6. Mixed Review 7. 8.

13 Chapter 3. Write an Equation Given Two Points 4. The line containing the points (3, 5) and ( 3, 0). 5. The line containing the points (10, 15) and (12, 20). 6. Mixed Review Translate into an algebraic sentence: One-third of a number is seven less than that number. 9. The perimeter of a square is 67 cm. What is the length of its side? 10. A hockey team played 17 games. They won two more than they lost. They lost 3 more than they tied. How many games did they win, lose, and tie? 11. Simplify ( ) (21 3) What is the opposite of 16.76? 13. Graph the following on a number line: 6, 11 3, 5.65, Simplify: [( )+(18 13 )+( 3.3)]. 13

LINEAR EQUATIONS IN TWO VARIABLES

LINEAR EQUATIONS IN TWO VARIABLES What You Should Learn Use slope to graph linear equations in two " variables. Find the slope of a line given two points on the line. Write linear equations in two variables.