3.5 Write and Graph Equations

Transcription

1 .5 Write and Graph Equations of Lines Goal p Find equations of lines. Your Notes VOCABULARY Slope-intercept form Standard form Eample Write an equation of a line from a graph Write an equation of the line in slope-intercept form. (0, ) (2, 2) Step Find the slope. Choose two points on the graph of the line, (0, ) and (2, 2). m Step 2 Find the -intercept. The line intersects the -ais at the point, so the -intercept is. Step Write the equation. 5 m b Use slope-intercept form. 5 Substitute for m and for b. 76 Lesson.5 Geometr Notetaking Guide Copright Holt McDougal. All rights reserved.

2 .5 Write and Graph Equations of Lines Goal p Find equations of lines. Your Notes VOCABULARY Slope-intercept form The general form of a linear equation in slope-intercept form is 5 m b, where m is the slope and b is the -intercept. Standard form The general form of a linear equation in standard form is A B 5 C, where A and B are not both zero. Eample Write an equation of a line from a graph Write an equation of the line in slope-intercept form. (0, ) (2, 2) Step Find the slope. Choose two points on the graph of the line, (0, ) and (2, 2). m 5 2 (2) } } Step 2 Find the -intercept. The line intersects the -ais at the point (0, ), so the -intercept is. Step Write the equation. 5 m b Use slope-intercept form Substitute 22 for m and for b. 76 Lesson.5 Geometr Notetaking Guide Copright Holt McDougal. All rights reserved.

3 The graph of a linear equation represents all the solutions of the equation. So, the given point must be a solution of the equation. Eample 2 Write an equation of the line passing through the point (, 2) that is parallel to the line with the equation Write an equation of a parallel line Step Find the slope m. The slope of a line parallel to is the same as the given line, so the slope is. Step 2 Find the -intercept b b using m 5 (, ) 5. 5 m b and Use slope-intercept form. 5 ( ) b Substitute for,, and m. 5 b Solve for b. Because m 5 and b 5, an equation of the line is 5. Checkpoint Complete the following eercises.. Write an equation of the line in the graph at the right. (2, ) (0, 25) 2. Write an equation of the line that passes through the point (22, 5) and is parallel to the line with the equation Copright Holt McDougal. All rights reserved. Lesson.5 Geometr Notetaking Guide 77

4 The graph of a linear equation represents all the solutions of the equation. So, the given point must be a solution of the equation. Eample 2 Write an equation of the line passing through the point (, 2) that is parallel to the line with the equation Write an equation of a parallel line Step Find the slope m. The slope of a line parallel to is the same as the given line, so the slope is 2. Step 2 Find the -intercept b b using m 5 2 and (, ) 5 (, 2). 5 m b Use slope-intercept form ( ) b Substitute for,, and m. 2 5 b Solve for b. Because m 5 2 and b 5 2, an equation of the line is Checkpoint Complete the following eercises.. Write an equation of the line in the graph at the right (2, ) (0, 25) 2. Write an equation of the line that passes through the point (22, 5) and is parallel to the line with the equation Copright Holt McDougal. All rights reserved. Lesson.5 Geometr Notetaking Guide 77

5 Eample Write an equation of a perpendicular line Write an equation of the line j passing through the point (, 2) that is perpendicular to the line k with the equation 5 2. Step Find the slope m of line j. The slope of k is. p m 5 The product of the slopes of perpendicular lines is. m 5 Divide each side b. Step 2 Find the -intercept b b using m 5 and (, ) 5. 5 m b Use slope-intercept form. 5 ( ) b Substitute for,, and m. 5 b Solve for b. Because m 5 and b 5, an equation of line j is 5. You can check that the lines j and k are perpendicular b graphing, then using a protractor to measure one of the angles formed b the lines. Checkpoint Complete the following eercise.. Write an equation of the line passing through the point (28, 22) that is perpendicular to the line with the equation Lesson.5 Geometr Notetaking Guide Copright Holt McDougal. All rights reserved.

6 Eample Write an equation of a perpendicular line Write an equation of the line j passing through the point (, 2) that is perpendicular to the line k with the equation 5 2. Step Find the slope m of line j. The slope of k is 2. 2 p m 5 2 The product of the slopes of perpendicular lines is 2. m 5 } Divide each side b 2. Step 2 Find the -intercept b b using m 5 } and (, ) 5 (, 2). 5 m b 2 5 } Use slope-intercept form. ( ) b Substitute for,, and m. 5 b Solve for b. Because m 5 } and b 5, an equation of line j is 5 }. You can check that the lines j and k are perpendicular b graphing, then using a protractor to measure one of the angles formed b the lines. k j (, 2) Checkpoint Complete the following eercise.. Write an equation of the line passing through the point (28, 22) that is perpendicular to the line with the equation } Lesson.5 Geometr Notetaking Guide Copright Holt McDougal. All rights reserved.

7 Eample 4 Rent The graph models the total cost of renting an apartment. Write an equation of the line. Eplain the meaning of the slope and the -intercept of the line. Step Find the slope. m 5 Write an equation of a line from a graph Cost (dollars) Renting Costs (5, 275) (2, 250) Time (months) 5 5 Step 2 Find the -intercept. Use a point on the graph. 5 m b Use slope-intercept form. 5 p b Substitute. 5 b Simplif. Step Write the equation. Because m 5 and b 5, an equation is 5. The equation 5 models the cost. The slope is the, and the is the initial cost to rent the apartment. Eample 5 Graph a line with equation in standard form Graph The equation is in standard form, so use the. Step Find the intercepts. To find the -intercept, To find the -intercept, let 5. let ( ) 5 6 2( ) Step 2 Graph the line. The line intersects the aes at and. Graph these points, then draw a line through the points. Copright Holt McDougal. All rights reserved. Lesson.5 Geometr Notetaking Guide 79

8 Eample 4 Write an equation of a line from a graph Rent The graph models the total cost of renting an apartment. Renting Costs Write an equation of the line. 000 Eplain the meaning of the slope 2500 (5, 275) and the -intercept of the line Step Find the slope. 000 (2, 250) m 5 }} Time (months) 25 5 } 5 75 Step 2 Find the -intercept. Use a point on the graph. 5 m b Use slope-intercept form p 2 b Substitute b Simplif. Step Write the equation. Because m 5 75 and b 5 500, an equation is The equation models the cost. The slope is the monthl rent, and the -intercept is the initial cost to rent the apartment. Cost (dollars) Eample 5 Graph a line with equation in standard form Graph The equation is in standard form, so use the intercepts. Step Find the intercepts. To find the -intercept, To find the -intercept, let 5 0. let ( 0 ) 5 6 2( 0 ) Step 2 Graph the line. The line intersects the aes at (0, 2) (, 0) and (0, 2). Graph these points, then draw a line through the points. (, 0) Copright Holt McDougal. All rights reserved. Lesson.5 Geometr Notetaking Guide 79

9 The point at which the costs are the same is sometimes called the breakeven point. Eample 6 Solve a real-world problem Subscriptions You can bu a magazine at a store for $. You can subscribe earl to the magazine for a flat fee of $8. After how man magazines is the subscription a better bu? Step Model each purchase with an equation. Cost of earl subscription: 5 Cost of one magazine: 5, where represents the number of magazines Step 2 Graph each equation. The point of intersection is. Using the graph, ou can see that it is cheaper to bu magazines individuall if ou bu less than magazines per ear. If ou bu more than magazines per ear, it is cheaper to bu a subscription. Cost (dollars) Magazine Purchases Number of magazines Checkpoint Complete the following eercises. 4. The equation models the total cost of joining a health club for ears. What are the meaning of the slope and -intercept of the line? 5. Graph 5 and 5. Homework 6. In Eample 6, suppose ou can bu the magazine at a different store for $2.50. After how man magazines is the subscription the better bu? 80 Lesson.5 Geometr Notetaking Guide Copright Holt McDougal. All rights reserved.

10 Eample 6 Solve a real-world problem Subscriptions You can bu a magazine at a store for $. You can subscribe earl to the magazine for a flat fee of $8. After how man magazines is the subscription a better bu? Step Model each purchase with an equation. Cost of earl subscription: 5 8 Cost of one magazine: 5, where represents the number of magazines The point at which the costs are the same is sometimes called the breakeven point. Step 2 Graph each equation. The point of intersection is (6, 8). Using the graph, ou can see that it is cheaper to bu magazines individuall if ou bu less than 6 magazines per ear. If ou bu more than 6 magazines per ear, it is cheaper to bu a subscription. Cost (dollars) Magazine Purchases (6, 8) Number of magazines Checkpoint Complete the following eercises. 4. The equation models the total cost of joining a health club for ears. What are the meaning of the slope and -intercept of the line? The slope is the cost per ear, $650, and the -intercept is the initiation fee, $ Graph 5 and 5. (0, ) 5 Homework (, 0) 5 6. In Eample 6, suppose ou can bu the magazine at a different store for $2.50. After how man magazines is the subscription the better bu? after 7 magazines 80 Lesson.5 Geometr Notetaking Guide Copright Holt McDougal. All rights reserved.