4.3 Graphing Using Intercepts Objective: Graph a linear equation using intercepts

What are intercepts? x-intercept y-intercept

Substitute 0 for y. Write original equation. To find the x- intercept, substitute 0 for y and solve for x. SOLUTION Find the x- intercept and the y- intercept of the graph of 2x + 7y = 28. Find the intercepts of the graph of an equation EXAMPLE 1 Solve for x. 2x + 7(0) = 28 x == x + 7y = 28

2(0) + 7y = 28 Find the intercepts of the graph of an equation EXAMPLE 1 To find the y- intercept, substitute 0 for x and solve for y. Write original equation. Substitute 0 for x. Solve for y. ANSWER The x- intercept is 14. The y- intercept is 4. 2x +7y = 28 y = 28 7 = 4

Examples 3x – 3y = 9 x – 4y = 18

Find the x- intercept and the y- intercept of the graph of the equation. GUIDED PRACTICE for Example x + 2y = 6 ANSWER 2, 3

GUIDED PRACTICE for Example x – 2y = 10 Find the x- intercept and the y- intercept of the graph of the equation. ANSWER 2.5, –5

GUIDED PRACTICE for Example 1 3. –3x + 5y = –15 Find the x- intercept and the y- intercept of the graph of the equation. ANSWER 5, –3

SOLUTION STEP 1 Use intercepts to graph an equation EXAMPLE 2 Graph the equation x + 2y = 4. x + 2y = 4 x =  x- intercept 4 Find the intercepts. x + 2(0) = y = 4 y =  y- intercept 2 x + 2y = 4

Use intercepts to graph an equation EXAMPLE 2 STEP 2 Plot points. The x- intercept is 4, so plot the point (4, 0). The y- intercept is 2, so plot the point (0, 2). Draw a line through the points.

Use a graph to find intercepts EXAMPLE 3 The graph crosses the x- axis at (2, 0). The x- intercept is 2. The graph crosses the y- axis at (0, –1). The y- intercept is –1.

Use intercepts to graph an equation EXAMPLE 2 GUIDED PRACTICE for Examples 2 and 3 Graph the equation 6x + 7y = 42. Label the points where the line crosses the axes. 4. ANSWER

GUIDED PRACTICE for Examples 2 and 3 Identify the x- intercept and the y- intercept of the graph shown at the right. 5. –4, 2 ANSWER

Additional Examples Find the x-intercept and y-intercept and graph the equations 1.y = x – 2 2.y = x 3.y = ½x + ¼

EVENT PLANNING Solve a multi-step problem EXAMPLE 4 You are helping to plan an awards banquet for your school, and you need to rent tables to seat 180 people. Tables come in two sizes. Small tables seat 4 people, and large tables seat 6 people. This situation can be modeled by the equation 4x + 6y = 180 where x is the number of small tables and y is the number of large tables. Find the intercepts of the graph of the equation.

SOLUTION STEP 1 Solve a multi-step problem EXAMPLE 4 Give four possibilities for the number of each size table you could rent. Graph the equation. Find the intercepts. 4x + 6(0) = 180 x =  x -intercept 45 4x + 6y =180 4(0) + 6y = 180 y =  y -intercept 30 4x + 6y = 180

Solve a multi-step problem EXAMPLE 4 Since x and y both represent numbers of tables, neither x nor y can be negative. So, instead of drawing a line, draw the part of the line that is in Quadrant I. STEP 2 Graph the equation. The x- intercept is 45, so plot the point (45, 0). The y- intercept is 30, so plot the point (0, 30).

Solve a multi-step problem EXAMPLE 4 STEP 3 Find the number of tables. For this problem, only whole - number values of x and y make sense. You can see that the line passes through the points (0, 30), (15, 20), (30, 10), and (45, 0).

Solve a multi-step problem EXAMPLE 4 So, four possible combinations of tables that will seat 180 people are : 0 small and 30 large, 15 small and 20 large, 30 small and 10 large,and 45 small and 0 large.

GUIDED PRACTICE EXAMPLE 2 GUIDED PRACTICE for Example 4 6. WHAT IF? In Example 4, suppose the small tables cost $9 to rent and the large tables cost $14. Of the four possible combinations of tables given in the example, which rental is the least expensive? Explain. 45 small tables and no large tables; if you rent 45 small tables it costs $405, all other combinations are more expensive. ANSWER

SUBMERSIBLES Use a linear model EXAMPLE 5 A submersible designed to explore the ocean floor is at an elevation of –13,000 feet (13,000 feet below sea level ). The submersible ascends to the surface at an average rate of 650 feet per minute. The elevation e ( in feet ) of the submersible is given by the function e = 650t – 13,000 where t is the time ( in minutes ) since the submersible began to ascend.

SOLUTION Use a linear model EXAMPLE 5 STEP 1 13,000 =650t 650t – 13,0000 = Find the intercepts. = t= t20  t -intercept = –13,000 e  e- intercept e = 650(0) – 13,000 Graph the function and identify its domain and range. Find the intercepts of the graph of the function and state what the intercepts represent.

Use a linear model EXAMPLE 5 The t- intercept represents the number of minutes the submersible takes to reach an elevation of 0 feet ( sea level ). The e- intercept represents the elevation of the submersible after 0 minutes ( the time the ascent begins ).

Use a linear model EXAMPLE 5 Graph the function using the intercepts. The submersible starts at an elevation of –13,000 feet and ascends to an elevation of 0 feet. So, the range of the function is –13,000 ≤ e ≤ 0. From the graph, you can see that the domain of the function is 0 ≤ t ≤ 20. STEP 2

Use a linear model EXAMPLE 5 GUIDED PRACTICE for Example 5 WHAT IF? in example 5, suppose the elevation of a second submersible is given by e = 500t – 10,000. Graph the function and identify its domain and range. 7. domain: 0 ≤ t ≤ 20, range: –10,000 ≤ e ≤ 0 ANSWER

Homework Starting on page 229: 1 - 3, 4 – 26 evens, 28 – 30, 34 – 40, 44 – 48