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Linear Functions and Equations Exploration

Learn to write linear functions using slope-intercept and point-slope forms, apply them to solve problems, and identify slopes and intercepts. Practice finding slopes and writing equations in slope-intercept form.

douglasjohnson
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Linear Functions and Equations Exploration

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  1. Objectives Use slope-intercept form and point-slope form to write linear functions. Write linear functions to solve problems. Recall from Lesson 2-3 that the slope-intercept form of a linear equation is y=mx + b, where m is the slope of the line and b is its y-intercept.

  2. m = – and b = 1. 3 4 The equation of the line is y = – x + 1. 3 4 Slope is = = – . rise run y = –x + 1 3 4 3 4 –3 4 Write the equation of the graphed line in slope-intercept form. Identify the y-intercept. The y-intercept b is 1. Step 1 Step 2 Find the slope. Write the equation in slope-intercept form. Step 3 y = mx + b

  3. Write the equation of the graphed line in slope-intercept form.

  4. Notice that for two points on a line, the rise is the differences in the y-coordinates, and the run is the differences in the x-coordinates. Using this information, we can define the slope of a line by using a formula.

  5. Find the slope of the line through (–1, 1) and (2, –5). Let (x1, y1) be (–1, 1) and (x2, y2) be (2, –5). The slope of the line is –2.

  6. 3 4 The slope of the line is . Find the slope of the line. Let (x1, y1) be (4, 2) and (x2, y2) be (8, 5).

  7. Find the slope of the line. Find the slope of the line shown. Find the slope of the line through (2,–5) and (–3, –5).

  8. Because the slope of line is constant, it is possible to use any point on a line and the slope of the line to write an equation of the line in point-slope form.

  9. In slope-intercept form, write the equation of the line that contains the points in the table. First, find the slope. Next, choose a point, and use either form of the equation of a line. Method A Point-Slope Form Method B Slope-intercept Form

  10. Write the equation of the line in slope-intercept form with slope –5 through (1, 3). Method A Point-Slope Form y – y1= m(x – x1) y – (3)= –5(x – 1) Substitute. y – 3 = –5(x – 1) Simplify. Rewrite in slope-intercept form. y – 3 = –5(x – 1) y – 3 = –5x + 5 Distribute. The equation of the slope is y = –5x + 8. y = –5x + 8 Solve for y.

  11. Write the equation of the line in slope-intercept form through (–2, –3) and (2, 5). First, find the slope. Let (x1, y1) be (–2,–3) and (x2, y2) be (2, 5). Method B Slope-Intercept Form y= mx+ b Rewrite the equation using m and b. 5= (2)2+ b 5 = 4 + b y =2x+ 1 y = mx + b 1 = b The equation of the line is y = 2x + 1.

  12. The table shows the rents and selling prices of properties from a game. Express the rent as a function of the selling price. Let x = selling price and y = rent. Find the slope by choosing two points. Let (x1, y1) be (75, 9)and (x2, y2) be (90, 12).

  13. Example 4A Continued To find the equation for the rent function, use point-slope form. y – y1= m(x – x1) Use the data in the first row of the table. Simplify.

  14. Write the equation of the line in slope-intercept form. parallel toy = 1.8x + 3 and through (5, 2) m = 1.8 y –2 = 1.8(x – 5) y –2 = 1.8x – 9 y= 1.8x – 7

  15. The slope of the given line is , so the slope of the perpendicular line is the opposite reciprocal, . Write the equation of the line in slope-intercept form. Perpendicular to and through (9, –2)

  16. Write the equation of the line in slope-intercept form. parallel to y = 5x – 3 and through (1, 4) perpendicular to y = ⅜x – 2 and through (3, 1)

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