Q: Which of the following is the correct quadratic formula for finding the $x$-intercepts of $ax^2 + bx + c = 0$?

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: If the discriminant $b^2 - 4ac$ of a quadratic equation is zero, what does this tell you about the graph?

The graph touches the $x$-axis at one point (one real root).

Question 1

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

Accepted Answer

$(3, 8)$

Question 2

Which of the following best describes the axis of symmetry for the quadratic function $f(x) = 4x^2 - 8x + 5$?

Accepted Answer

$x = 1$

Question 3

A ball is thrown upward from a platform 20 meters high with an initial velocity of 12 m/s. Its height after $t$ seconds is given by $h(t) = -4.9t^2 + 12t + 20$. What is the maximum height the ball reaches?

Accepted Answer

$27.35$ meters

Question 4

What is the discriminant of the quadratic equation $2x^2 - 4x + 3 = 0$ and how many real $x$-intercepts does the graph have?

Accepted Answer

Discriminant $= -8$, no real $x$-intercepts

Question 5

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

Accepted Answer

$(x-2)(x-3)$

Question 6

A rectangular garden is to be fenced on three sides (the fourth side is against a wall) using 100 meters of fencing. If the length parallel to the wall is $x$, what value of $x$ maximizes the area?

Accepted Answer

$50$ meters

Question 7

Given $f(x) = x^2 + 6x + 8$, what are the $x$-intercepts?

Accepted Answer

$x = -4$ and $x = -2$

Question 8

Which of the following statements is true about the graph of $f(x) = -3x^2 + 12x - 7$?

Accepted Answer

It opens downward and has a maximum value.

Question 9

If a quadratic function has a vertex at $(2, -5)$ and passes through the point $(4, 3)$, which of the following could be its equation?

Question 10

Which of the following is the range of the function $f(x) = -x^2 + 4x - 1$?

Accepted Answer

$(-\infty, 3]$

Question 11

A quadratic function $f(x) = ax^2 + bx + c$ has its vertex at $(1, -4)$ and passes through $(3, 0)$. If $a = 1$, what is the value of $b$?

Accepted Answer

$b = -6$

Question 12

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

Accepted Answer

If $|a| > 1$, the parabola is narrower; if $0 < |a| < 1$, it is wider.

Question 13

A projectile is launched from a height of 50 meters with an initial velocity of 20 m/s. Its height after $t$ seconds is $h(t) = -4.9t^2 + 20t + 50$. How long does it take to hit the ground?

Accepted Answer

$t \approx 5.23$ seconds

Question 14

Given $f(x) = (x+1)^2 - 9$, what are the $x$-intercepts?

Accepted Answer

$x = 2$ and $x = -4$

Question 15

Which of the following is the correct quadratic formula for finding the $x$-intercepts of $ax^2 + bx + c = 0$?

Accepted Answer

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Question 16

If the discriminant $b^2 - 4ac$ of a quadratic equation is zero, what does this tell you about the graph?

Accepted Answer

The graph touches the $x$-axis at one point (one real root).

Question 17

A box with no top is to be made from a $20$ cm by $20$ cm square by cutting equal squares from each corner and folding up the sides. If the volume must be $200$ cm$^3$, what size square should be cut from each corner?

Accepted Answer

$x = 2$ cm

Question 18

Given $f(x) = -x^2 + 6x - 5$, what is the maximum value of $f(x)$?

Accepted Answer

$4$

Question 19

Which of the following is the domain of any quadratic function $f(x) = ax^2 + bx + c$?

Accepted Answer

$(-\infty, \infty)$

D2-D4

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