5.2E Exercises

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Identifying Quadratic Functions

Tell whether the functions are quadratic functions, and if necessary, write them in standard form.

$ f(x) = 3x^2 $
$ g(x) = 2x^{\frac{3}{2}} - x $
$ h(x) = \frac{1}{x^2} + 3 $
$ p(x) = (x+1)(x-2) $
$ q(x) = (x+1)^2(x-2) $
$ r(x) = 4 - (x+1)^2 $

Answer

Yes.
No.
No.
Yes, $ p(x) = x^2 - x - 2 $.
No, when expanded, this will have an $x^3$ term.
Yes, $ r(x) = -x^2 -2x + 3 $.

Standard vs. Vertex Form

If the quadratic function is in standard form, convert it to vertex form and identify the vertex $ (h,k)$. If it is already in vertex form, convert it to standard form and identify the constants $a, b,$ and $c$.

$ f(x) = -2(x+1)^2 + 5 $
$ f(x) = (x+2)^2 - 13 $
$ g(x) = x^2 + 14x + 39 $
$ g(x) = -4x^2 - 16x - 21 $

Answer

$ f(x) = -2x^2 - 4x + 3 $, $a = -2, b = -4, c = 3 $.
$ f(x) = x^2 + 4x - 9 $, $ a = 1, b = 4, c = -9 $.
$ g(x) = (x+7)^2 - 10 $, $ (h,k) = (-7, -10) $.
$ g(x) = -4(x+2)^2 - 5 $, $ (h,k) = (-2, -5) $.

Domain and Range

Give the domain and range of each quadratic function.

$ q(x) = x^2 $
$ f(x) = 4 - (x+1)^2 $
$ g(x) = -2(x+1)^2 + 5 $
$ p(x) = 3x^2 $
$ h(x) = (x+2)^2 - 13 $
$ r(x) = -x^2 + 4 $

Answer

Domain: $ \mathbb{R} $. Range: $ [0, \infty) $.
Domain: $ \mathbb{R} $. Range: $ (-\infty, 4] $.
Domain: $ \mathbb{R} $. Range: $ (-\infty, 5] $.
Domain: $ \mathbb{R} $. Range: $ [0, \infty) $.
Domain: $ \mathbb{R} $. Range: $ [-13, \infty) $.
Domain: $ \mathbb{R} $. Range: $ (-\infty, 4] $.

Finding Real Roots

Find any and all real roots of the quadratic functions. Classify the scenario: (I) one repeated real root, (II) two distinct real roots, or (III) no real roots.

$ p(x) = x^2 - x - 2 $
$ q(x) = x^2 $
$ r(x) = x^2 - 6x + 9 $
$ f(x) = 2x^2 + x + 4 $
$ h(x) = 3x^2 +x + 12 $
$ g(x) = 2x^2 - x - 1 $

Answer

$x = -1, 2 $ (II)
$ x = 0, 0 $ (I)
$ x = 3, 3 $ (I)
No real roots (III)
No real roots (III)
$ x = -\frac{1}{2}, 1 $ (II)

Sketching Graphs of Quadratic Functions

Sketch the graphs of the functions (without needing technology).

$ f(x) = (x-5)^2 - 1 $
$ g(x) = -x^2 - 3 $
$ h(x) = x^2 + 2x - 15 $

Answer

1.	2.	3.
$hw1.png$	$hw2.png$	$hw3.png$

Parabolic Motion

When a ball is launched straight upward from the ground with an initial velocity $v_0$ ft/s, its height in feet after $t$ seconds is given by $h(t) = v_0 t - 16t^2$. Let $v_0$ be 32 ft/s and convert this function into vertex form. What is the maximum height achieved by the ball, and when does it begin to fall down? When will it hit the ground again?

Repeat the process for $v_0 = 160$ ft/s.

Answer

For $v_0 = 32$ ft/s: $h(t) = -16(t-1)^2 + 16 $. Max height: 16ft, reached at time $t = 1$s. It hits the ground after 2s.

For $v_0 = 160$ ft/s: $h(t) = -16(t-5)^2 + 400$. Max height: 400ft, reached at time $t = 5$s. It hits the ground after 10s.

Maximizing Profits

If my monthly revenue function (money coming in) at De Wolf's De Washing Machines is given by $R(x) = 1200x $, where $x$ is the number of machines I sell, and my monthly cost function (money going out) is given by $C(x) = x^2 + 400x $, find the quadratic function that calculates my profits. What will the profit be if 100 machines sell? How many machines sold would result in a profit of $120,000? Find the maximum possible profit and the number of machines sold to achieve it.

Answer

$ P(x) = R(x) - C(x) = 800x - x^2 $. If 100 machines sell, profit is $70,000. Profit will be $120,000 if $x = 200$ or $600$.

Converted to vertex form: $ P(x) = -(x-400)^2 + 160000 $. Max profit: $160,000, when $x = 400$.

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