Question 1

Given the quadratic function $f(x) = 2x^2 - 8x + 3$, what are the coordinates of the vertex?

Accepted Answer

$(2, -5)$

Question 2

A ball is thrown upward from the ground with an initial velocity such that its height in meters after $t$ seconds is given by $h(t) = -5t^2 + 20t$. What is the maximum height reached by the ball?

Accepted Answer

$25$ meters

Question 3

Which of the following is the correct formula to find the $x$-coordinate of the vertex of a parabola given by $f(x) = ax^2 + bx + c$?

Accepted Answer

$x = -\frac{b}{2a}$

Question 4

A company models its profit $P(x)$ in thousands of dollars by the function $P(x) = -3x^2 + 12x + 5$, where $x$ is the number of units sold (in thousands). How many units should be sold to maximize profit?

Accepted Answer

$x = 2$

Question 5

If the quadratic function $f(x) = x^2 - 6x + 8$ is rewritten in vertex form, what is the vertex?

Accepted Answer

$(3, -1)$

Question 6

A satellite dish is shaped like a parabola. Its cross-section can be modeled by $y = 0.5x^2 - 4x + 7$. What is the minimum value of $y$?

Accepted Answer

$3$

Question 7

Given $f(x) = -x^2 + 4x - 1$, what is the axis of symmetry of the parabola?

Accepted Answer

$x = 2$

Question 8

A projectile is launched and its height in meters after $t$ seconds is given by $h(t) = -4.9t^2 + 14.7t + 2$. After how many seconds does it reach its maximum height?

Accepted Answer

$t = 1.5$

Question 9

Which of the following statements is true about the graph of $f(x) = ax^2 + bx + c$ if $a > 0$?

Accepted Answer

The parabola opens upward and has a minimum point.

Question 10

A rectangular garden is to be fenced on three sides with 60 meters of fencing, with one side against a wall. What is the maximum area that can be enclosed?

Accepted Answer

$300$ square meters

Question 11

If $f(x) = 3x^2 - 12x + 7$, what is the minimum value of $f(x)$?

Accepted Answer

$-5$

Question 12

A quadratic function models the path of a diver: $h(x) = -2x^2 + 8x + 3$, where $h$ is the height in meters and $x$ is the horizontal distance in meters. What is the maximum height the diver reaches?

Accepted Answer

$11$ meters

Question 13

Which of the following is the correct vertex form of the quadratic $f(x) = x^2 + 6x + 5$?

Accepted Answer

$f(x) = (x + 3)^2 - 4$

Question 14

A farmer wants to build a rectangular pen against a barn using 80 meters of fencing for the other three sides. What is the maximum area the pen can have?

Accepted Answer

$800$ square meters

Question 15

Given the quadratic function $f(x) = -2x^2 + 4x + 6$, what is the $y$-coordinate of the vertex?

Accepted Answer

$10$

Question 16

If the vertex of a parabola is at $(h, k)$ and the equation is $f(x) = a(x - h)^2 + k$, which value represents the minimum or maximum value of the function?

Accepted Answer

$k$

Question 17

A quadratic function $f(x) = ax^2 + bx + c$ has its vertex at $(1, -4)$. If $a = 2$ and $b = -4$, what is the value of $c$?

Accepted Answer

$-2$

Question 18

A bridge is designed in the shape of a parabola with its vertex at the highest point. If the equation is $y = -0.25x^2 + 6$, what is the height of the bridge at its vertex?

Accepted Answer

$6$ meters

Question 19

Which of the following best describes the effect of increasing the value of $a$ in $f(x) = a(x - h)^2 + k$ (with $a > 0$) on the graph of the parabola?

Accepted Answer

The parabola becomes narrower.

Question 20

A quadratic function $f(x) = x^2 + bx + c$ has its vertex at $(2, -3)$. What is the value of $b$?

Accepted Answer

$-4$

Quadratic Functions and Parabolas: Vertex and Applications

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