9.6 Graph Quadratic Functions Using Properties

Solve: $2 x^{2} + 3 x - 2 = 0 .$
If you missed this problem, review the reference.

Evaluate $- \frac{b}{2 a}$ when a = 3 and b = −6.
If you missed this problem, review the reference.

Determine whether the graph of each function is a parabola that opens upward or downward:

ⓐ $f \left(x\right) = 2 x^{2} + 5 x - 2$ ⓑ $f \left(x\right) = −3 x^{2} - 4 x + 7 .$

Determine whether the graph of each function is a parabola that opens upward or downward:

ⓐ $f \left(x\right) = −2 x^{2} - 2 x - 3$ ⓑ $f \left(x\right) = 5 x^{2} - 2 x - 1 .$

For the graph of $f \left(x\right) = 2 x^{2} - 8 x + 1$ find:

ⓐ the axis of symmetry ⓑ the vertex.

For the graph of $f \left(x\right) = 2 x^{2} - 4 x - 3$ find:

ⓐ the axis of symmetry ⓑ the vertex.

Find the intercepts of the parabola whose function is $f \left(x\right) = x^{2} + 2 x - 8 .$

Find the intercepts of the parabola whose function is $f \left(x\right) = x^{2} - 4 x - 12 .$

Find the intercepts of the parabola whose function is $f \left(x\right) = 3 x^{2} + 4 x + 4 .$

Find the intercepts of the parabola whose function is $f \left(x\right) = x^{2} - 4 x - 5 .$

Find the maximum or minimum value of the quadratic function $f \left(x\right) = x^{2} - 8 x + 12 .$

Find the maximum or minimum value of the quadratic function $f \left(x\right) = −4 x^{2} + 16 x - 11 .$

The quadratic equation h(t) = −16t² + 176t + 4 models the height of a volleyball hit straight upwards with velocity 176 feet per second from a height of 4 feet.

ⓐ How many seconds will it take the volleyball to reach its maximum height? ⓑ Find the maximum height of the volleyball.

Solve, rounding answers to the nearest tenth.

The quadratic function h(t) = −16t² + 128t + 32 is used to find the height of a stone thrown upward from a height of 32 feet at a rate of 128 ft/sec. How long will it take for the stone to reach its maximum height? What is the maximum height?

A path of a toy rocket thrown upward from the ground at a rate of 208 ft/sec is modeled by the quadratic function of h(t) = −16t² + 208t. When will the rocket reach its maximum height? What will be the maximum height?

For each of the following exercises, determine if the parabola opens up or down.

ⓐ $f \left(x\right) = −2 x^{2} - 6 x - 7$ ⓑ $f \left(x\right) = 6 x^{2} + 2 x + 3$

For each of the following exercises, determine if the parabola opens up or down.

ⓐ $f \left(x\right) = −3 x^{2} + 5 x - 1$ ⓑ $f \left(x\right) = 2 x^{2} - 4 x + 5$

In the following functions, find ⓐ the equation of the axis of symmetry and ⓑ the vertex of its graph.

$f \left(x\right) = x^{2} + 8 x - 1$

In the following functions, find ⓐ the equation of the axis of symmetry and ⓑ the vertex of its graph.

$f \left(x\right) = − x^{2} + 2 x + 5$

Find the intercepts of the parabola whose function is given.

$f \left(x\right) = x^{2} + 7 x + 6$

Find the intercepts of the parabola whose function is given.

$f \left(x\right) = x^{2} + 8 x + 12$

Find the intercepts of the parabola whose function is given.

$f \left(x\right) = − x^{2} + 8 x - 19$

Find the intercepts of the parabola whose function is given.

$f \left(x\right) = x^{2} + 6 x + 13$

Find the intercepts of the parabola whose function is given.

$f \left(x\right) = 4 x^{2} - 20 x + 25$

Find the intercepts of the parabola whose function is given.

$f \left(x\right) = − x^{2} - 6 x - 9$

Find the maximum or minimum value of each function.

$f \left(x\right) = 2 x^{2} + x - 1$

Find the maximum or minimum value of each function.

$y = x^{2} - 6 x + 15$

Find the maximum or minimum value of each function.

$y = −9 x^{2} + 16$

Solve. Round answers to the nearest tenth.

An arrow is shot vertically upward from a platform 45 feet high at a rate of 168 ft/sec. Use the quadratic function h(t) = −16t² + 168t + 45 find how long it will take the arrow to reach its maximum height, and then find the maximum height.

Solve. Round answers to the nearest tenth.

A ball is thrown vertically upward from the ground with an initial velocity of 109 ft/sec. Use the quadratic function h(t) = −16t² + 109t + 0 to find how long it will take for the ball to reach its maximum height, and then find the maximum height.

Solve. Round answers to the nearest tenth.

A computer store owner estimates that by charging x dollars each for a certain computer, he can sell 40 − x computers each week. The quadratic function R(x) = −x² +40x is used to find the revenue, R, received when the selling price of a computer is x, Find the selling price that will give him the maximum revenue, and then find the amount of the maximum re…

Solve. Round answers to the nearest tenth.

A retailer who sells fashion boots estimates that by selling them for x dollars each, he will be able to sell 70 − x boots a week. Use the quadratic function R(x) = −x² +70x to find the revenue received when the average selling price of a pair of fashion boots is x. Find the selling price that will give him the maximum revenue, and then find the amount of…

Solve. Round answers to the nearest tenth.

A rancher is going to fence three sides of a corral next to a river. He needs to maximize the corral area using 240 feet of fencing. The quadratic equation $A \left(x\right) = x \left(120 - \frac{x}{2}\right)$ gives the area of the corral, A, for the length, x, of the corral along the river. Find the length of the corral along the river that will give the maximum area, and then find the maximum area of the…

Solve. Round answers to the nearest tenth.

A land owner is planning to build a fenced in rectangular patio behind his garage, using his garage as one of the “walls.” He wants to maximize the area using 80 feet of fencing. The quadratic function A(x) = x(80 − 2x) gives the area of the patio, where x is the width of one side. Find the maximum area of the patio.

How do the graphs of the functions $f \left(x\right) = x^{2}$ and $f \left(x\right) = x^{2} - 1$ differ? We graphed them at the start of this section. What is the difference between their graphs? How are their graphs the same?

Explain how to find the intercepts of a parabola.