Question 1

Which of the following describes the transformation of the graph of $y = f(x)$ to $y = f(x-4) + 2$?

Accepted Answer

Shift 4 units to the right and 2 units upward

Question 2

Given the function $y = -f(x)$, how is its graph related to the graph of $y = f(x)$?

Accepted Answer

It is reflected across the x-axis

Question 3

If $f(x) = x^2$, which of the following is the vertex of the parabola $g(x) = (x-3)^2 + 5$?

Accepted Answer

$(3, 5)$

Question 4

Which transformation results in a vertical stretch of the graph of $y = f(x)$ by a factor of 3?

Accepted Answer

$y = 3f(x)$

Question 5

Given $f(x) = x^2$, which equation represents a reflection of $f(x)$ across the y-axis?

Accepted Answer

$y = f(-x)$

Question 6

A company’s cost and revenue functions are $C(x) = 12x + 32$ and $R(x) = -4x^2 + 36x$. What is the minimum break-even quantity?

Accepted Answer

$x = 2$

Question 7

For the quadratic function $f(x) = ax^2 + bx + c$, the vertex $(h, k)$ can be found using which formulas?

Accepted Answer

$h = -\frac{b}{2a},\ k = f(h)$

Question 8

If $f(x) = x^2$, which of the following graphs is compressed vertically by a factor of $\frac{1}{2}$?

Accepted Answer

$y = \frac{1}{2}x^2$

Question 9

Given $f(x) = x^2$, which transformation results in the graph opening downward and shifted 2 units down?

Accepted Answer

$y = -x^2 - 2$

Question 10

Suppose the age at first marriage for women in the US is modeled by $P(t) = 0.011t^2 - 0.712t + 20.755$, where $t$ is years since 1900. What is the predicted age in 2025?

Accepted Answer

$29.2$

Question 11

Which of the following statements is true about the graph of $y = af(x)$ compared to $y = f(x)$?

Accepted Answer

It is stretched vertically if $|a| > 1$

Question 12

Given $f(x) = x^2 - 8x + 17$, rewrite it in vertex form and identify the vertex.

Accepted Answer

Vertex form: $f(x) = (x-4)^2 + 1$, Vertex: $(4, 1)$

Question 13

If $y = \sqrt{x-1}$, how is this graph related to $y = \sqrt{x}$?

Accepted Answer

Shifted 1 unit to the right

Question 14

A parabola has vertex $(-3, -8)$ and passes through $(0, 10)$. What is its equation in the form $f(x) = a(x-h)^2 + k$?

Accepted Answer

$f(x) = 2(x+3)^2 - 8$

Question 15

Given $C(x) = 12x + 32$ and $R(x) = -4x^2 + 36x$, what is the maximum profit and at what quantity does it occur?

Accepted Answer

Maximum profit is $4,000$ at $x = 3$

Question 16

Which of the following best describes the effect of $y = f(-x)$ on the graph of $y = f(x)$?

Accepted Answer

Reflection across the y-axis

Question 17

If $f(x) = x^2$, which transformation results in the graph being shifted 5 units upward and reflected across the x-axis?

Accepted Answer

$y = -x^2 + 5$

Question 18

Given the quadratic function $f(x) = ax^2 + bx + c$, what is the axis of symmetry?

Accepted Answer

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

Question 19

If $y = f(x+2) - 3$, how is the graph of $y = f(x)$ transformed?

Accepted Answer

Shifted 2 units to the left and 3 units downward

Transformations of Graphs and Quadratic Functions in Business Calculus

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