Q: Which of the following statements is true about the graph of $f(x) = a(x - h)^2 + k$ when $a < 0$?

The parabola opens downward and has a maximum at $(h, k)$.

Q: A landscaper has 24 ft of stone to enclose three sides of a rectangular pond (the fourth side is a wall). What dimensions maximize the area?

Width $= 6$ ft, Length $= 12$ ft

Q: For the function $f(x) = -x^2 + 14x - 47$, what is the range?

$(-\infty, k]$ where $k$ is the maximum value at the vertex

Q: Which real-world scenario can be modeled by a quadratic function?

The height of a projectile over time after being launched upward.

Question 1

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

Accepted Answer

$(-3, -2)$

Question 2

For the function $f(x) = -2(x - 1)^2 + 3$, what is the axis of symmetry?

Accepted Answer

$x = 1$

Question 3

Which of the following statements is true about the graph of $f(x) = a(x - h)^2 + k$ when $a < 0$?

Accepted Answer

The parabola opens downward and has a maximum at $(h, k)$.

Question 4

Find the vertex of the quadratic function $f(x) = x^2 + 10x + 23$.

Accepted Answer

$(-5, -2)$

Question 5

A stone is thrown upward from a height of 30 ft with an initial velocity of 60 ft/sec. Its height after $t$ seconds is given by $h(t) = -16t^2 + 60t + 30$. At what time does the stone reach its maximum height?

Accepted Answer

$t = 1.875$ seconds

Question 6

Given $f(x) = ax^2 + bx + c$, what is the formula for the $x$-coordinate of the vertex?

Accepted Answer

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

Question 7

A landscaper has 24 ft of stone to enclose three sides of a rectangular pond (the fourth side is a wall). What dimensions maximize the area?

Accepted Answer

Width $= 6$ ft, Length $= 12$ ft

Question 8

For the function $f(x) = -x^2 + 14x - 47$, what is the range?

Accepted Answer

$(-\infty, k]$ where $k$ is the maximum value at the vertex

Question 9

If $f(x) = 3(x - 2)^2 + 5$, what is the minimum value of $f(x)$?

Accepted Answer

$5$

Question 10

Which of the following best describes the interval(s) where $f(x) = x^2 + 10x + 23$ is increasing?

Accepted Answer

$(-5, \infty)$

Question 11

Given $f(x) = -2(x + 4)^2 + 7$, what is the maximum value of $f(x)$?

Accepted Answer

$7$

Question 12

Which of the following is the correct vertex form for $f(x) = x^2 + 10x + 23$?

Accepted Answer

$f(x) = (x + 5)^2 - 2$

Question 13

A ball is thrown upward and its height is modeled by $h(t) = -16t^2 + 64t + 80$. What is the maximum height reached by the ball?

Accepted Answer

$144$ ft

Question 14

For $f(x) = a(x - h)^2 + k$, which parameter determines the direction the parabola opens?

Accepted Answer

$a$

Question 15

If the axis of symmetry of a parabola is $x = 4$ and the vertex is at $(4, -7)$, which could be the equation of the parabola?

Accepted Answer

$f(x) = 2(x - 4)^2 - 7$

Question 16

Which real-world scenario can be modeled by a quadratic function?

Accepted Answer

The height of a projectile over time after being launched upward.

Question 17

Given $f(x) = -x^2 + 14x - 47$, at what $x$-value does the maximum occur?

Accepted Answer

$x = 7$

Question 18

If $f(x) = x^2 - 6x + 8$, what is the minimum value of $f(x)$?

Accepted Answer

$-1$

Question 19

Which of the following best describes the effect of increasing $|a|$ in $f(x) = a(x - h)^2 + k$?

Accepted Answer

The parabola becomes narrower.

Analyzing Graphs of Quadratic Functions: Vertex, Axis of Symmetry, and Extrema

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