s olving q uadratic e quations u sing g raphs the solutions, or roots, of ax 2 + bx + c = 0 are the...

SOLVING QUADRATIC EQUATIONS USING GRAPHS

The solutions, or roots, of ax 2 + bx + c = 0 are the x-intercepts.

SOLVING QUADRATIC EQUATIONS GRAPH ICALLY

Write the equation in the form ax 2 + bx + c = 0.

Write the related function y = ax 2 + bx + c.

Sketch the graph of the function y = ax 2 + bx + c.

STEP 1

STEP 2

STEP 3

The solution of a quadratic equation in one variable x can be solved or checked graphically with the following steps:

Solve x 2 = 8 algebraically. 1

2 Check your solution graphically.

12

x 2 = 8

SOLUTION

Write original equation.

x 2 = 16 Multiply each side by 2.

Find the square root of each side.x = 4

Check these solutions using a graph.

Checking a Solution Using a Graph

CHECK

Checking a Solution Using a Graph

Write the equation in the form ax 2 + bx + c = 0

12

x 2 = 8 Rewrite original equation.

12

x 2 – 8 = 0 Subtract 8 from both sides.

y = 12

x2 – 8

Write the related function y = ax2 + bx + c.

1

2

Check these solutions using a graph.CHECK

Checking a Solution Using a Graph

2

3

Check these solutions using a graph.

Sketch graph of y =2

x2 – 8.1

The x-intercepts are 4, whichagrees with the algebraic solution.

Write the related function

y = 12

x2 – 8

y = ax2 + bx + c.

CHECK

4, 0– 4, 0

Solving an Equation Graphically

Solve x 2 – x = 2 graphically. Check your solution algebraically.

SOLUTION

Write the equation in the form ax 2 + bx + c = 0

x2 – x = 2 Write original equation.

x2 – x – 2 = 0 Subtract 2 from each side.

Write the related function y = ax2 + bx + c.

y = x2 – x – 2

1

2

Solving an Equation Graphically

Write the related function y = ax2 + bx + c.

y = x2 – x – 2

Sketch the graph of the function

y = x2 – x – 2

From the graph, the x-interceptsappear to be x = –1 and x = 2.

2

3

– 1, 0 2, 0

Solving an Equation Graphically

From the graph, the x-interceptsappear to be x = –1 and x = 2.

You can check this by substitution.

Check x = –1: Check x = 2:

x 2 – x = 2

(–1) 2 – (–1) 2=

?

1 + 1 = 2

x 2 – x = 2

4 – 2 = 2

2 2 – 2 = 2

?

– 1, 0 2, 0

CHECK

Comparing Two Quadratic Models

USING QUADRATIC MODELS IN REAL LIFE

A shot put champion performs an experiment for your math class. Assume that both times he releases the shot with the same initial speed but at different angles. The path of each put can be modeled by one of the equations below, where x represents the horizontal

distance (in feet) and y represents the height (in feet).

Initial angle of 35º: y = – 0.010735x 2 + 0.700208x + 6

Initial angle of 65º: y = – 0.040330x 2 + 2.144507x + 6

Which angle results in a farther throw?

Comparing Two Quadratic Models

SOLUTION

Begin by graphing both models in the same coordinate

plane. Use only positive x-values because x representsthe distance of the throw.

Which angle results in a farther throw?

Comparing Two Quadratic Models

PARABOLIC MOTION An angle of 45 º produces the maximum distance when an object is propelled from ground level. An angle

smaller than 45 º is better when the shot is released above the

ground. If the shot is released from 6 feet above the ground, a 43 º angle produces a maximum distance of 74.25 feet.

65º 45º

43º

35º