Question 1

Which of the following equations represents a parabola that opens upwards and has its vertex at the point $$(-2, 3)$$?

Accepted Answer

$$y = (x + 2)^2$ + 3$$

Question 2

Given the quadratic function $$f(x) = ax^2$ + bx + c$$, which value determines the number of distinct $$x-$$intercepts of the graph?

Accepted Answer

$$\Delta = b^2$ - 4ac$$

Question 3

A function $$f: D 	o \mathbb{R} is $$given. If $$f(x) = 0$$ for some $$x in$$ $$D$$, what does this mean graphically?

Accepted Answer

The graph of $$y = f(x)$$ passes through the $$x-$$axis at $$x$$

Question 4

Which of the following describes the effect of replacing $$x$$ with $$x - 3 in $$the function $$f(x)$$?

Accepted Answer

The graph shifts 3 units to the right

Question 5

Consider the cubic function $$f(x) = x^3$ - 5x^2$ + 3x - 2. $$Which statement about its graph is correct?

Accepted Answer

It can have up to three $$x-$$intercepts

Question 6

If the graph of $$y = f(x) is $$shifted up by 4 units, which equation represents the new graph?

Accepted Answer

$$y = f(x) + 4$$

Question 7

Which of the following is the equation of a line with slope $$-2$$ passing through the point $$(1, 5)$$?

Accepted Answer

$$y = -2x + 7$$

Question 8

For the function $$f(x) = x^2$, which of the following describes the graph of y = f(x) - 5$?

Accepted Answer

The graph of $$y = x^2$ shifted 5 units down

Question 9

A hospital monitors the temperature of a patient over time, modeled by $$T(t) = 37 + 0.5t - 0.02t^2$, where t$ is hours since admission. What is the maximum temperature reached?

Accepted Answer

$$38.25$$

Question 10

Which of the following statements about the graph of $$y = x^2$ is true?

Accepted Answer

It is symmetric about the $$y-$$axis

Question 11

The graph of $$y = f(x) is $$shifted 2 units to the left. Which equation represents the new graph?

Accepted Answer

$$y = f(x + 2)$$

Question 12

A quadratic function $$f(x) = ax^2$ + bx + c$$ has a discriminant $$\Delta = 0. $$What does this imply about its graph?

Accepted Answer

It touches the $$x-$$axis at exactly one point

Question 13

Which of the following is the graph of a constant function?

Accepted Answer

A horizontal line

Question 14

If $$f(x) = x^2$ and g(x) = f(x - 1) + 2$, what is the vertex of the graph of y = g(x$$)$$?

Accepted Answer

$$(1, 2)$$

Question 15

A chemical reaction's rate is modeled by $$R(t) = 3t^2$ - 12t + 9. At $$what time $$t$$ does the reaction rate reach its minimum?

Accepted Answer

$$t = 2$$

Question 16

Which of the following is true about the graph of $$y = x^3$?

Accepted Answer

It passes through the origin and is symmetric about the origin

Question 17

If the graph of $$y = f(x) is $$shifted 5 units down, which equation represents the new graph?

Accepted Answer

$$y = f(x) - 5$$

Question 18

A quadratic function $$f(x) = x^2$ - 4x + 3$$ models the height of a ball thrown upward, where $$x is $$time in seconds. At what time does the ball reach its maximum height?

Accepted Answer

$$x = 2$$

Question 19

Which of the following describes the graph of $$y = -x^2$ compared to y$$ = $$x^2$$?

Accepted Answer

It opens downward and is symmetric about the $$y-$$axis

Question 20

If $$f(x) = x^2$ and h(x) = f(x + 3) - 2$, what is the vertex of the graph of y = h(x$$)$$?

Accepted Answer

$$(-3, -2)$$

Introduction to Functions and Graphs in Calculus

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