Exercises: Graph exponential, logarithmic, and trigonometric functions - Common Core Math - HS Functions

A

Warm-Up: Review What You Know

These problems review skills you have already learned.

1.

Which of the following expressions is equivalent to $2^{-3}$ ?

A.

$-8$

B.

$\frac{1}{8}$

C.

$\frac{1}{6}$

D.

$-\frac{1}{8}$

2.

The graph of $f(x) = 3^x$ is shifted down 5 units. What is the equation of the transformed function?

A.

$f(x) = 3^{x-5}$

B.

$f(x) = 3^x + 5$

C.

$f(x) = 3^x - 5$

D.

$f(x) = 3^{x+5}$

3.

The parent function $y = \sin(x)$ has a maximum value of 1 and a minimum value of $-1$ .
What is the amplitude of the parent sine function?

A.

2

B.

$-1$

C.

1

D.

0

B

Fluency Practice

Apply the graphing procedures and identify key features.

C

Varied Practice

These problems present the same skills in different formats.

D

Word Problems

Apply your knowledge of function families to real-world contexts.

1.

A bacterial culture starts with 200 cells and doubles every hour.
The number of cells after $t$ hours is modeled by $f(t) = 200 \cdot 2^t$ .

The horizontal asymptote limits the model's output as $t \to -\infty$ .
Enter the $y$ -value of the horizontal asymptote.

2.

The loudness $L$ (in decibels) of a sound is modeled by
$L = 10 \cdot \log_{10}\!\left(\frac{I}{I_0}\right)$
where $I$ is the sound intensity and $I_0$ is the reference intensity.
A sound has intensity $I = 10{,}000 \cdot I_0$ .

What is the loudness $L$ in decibels?

$Graph of tidal depth h = 4 sin(pi/6 * t) + 10 over 24 hours showing midline at 10 ft, high tide 14 ft, low tide 6 ft, and period 12 hours$

3.

Ocean tides at a harbor can be modeled by the function
$h(t) = 4\sin\!\left(\frac{\pi}{6}t\right) + 10$
where $h$ is the water depth in feet and $t$ is the time in hours after midnight.

1.

What is the amplitude of the tidal function in feet?

2.

What is the period of the tidal function in hours?

E

Error Analysis

Each problem shows a student's work that contains an error. Identify the mistake.

1.

A student was asked to compare $f(x) = 2^x$ and $g(x) = x^2$ for large values of $x$ .
They wrote:

"Both functions grow the same way — they both curve upward faster and faster. By $x = 10$ , they are both very large, so they are the same type of function."

What error did the student make?

A.

The student is correct — exponential and polynomial functions are the same for large $x$ .

B.

The student confused the shape of the curves — $x^2$ curves downward, not upward.

C.

The student confused function families — exponential growth eventually exceeds any polynomial, no matter how large $x$ gets.

D.

The student used the wrong values — $2^{10}$ and $10^2$ are actually equal.

2.

A student graphed $f(x) = \log_2(x)$ and concluded:

"The graph clearly flattens out as $x$ increases. Since it appears to level off, the function has a horizontal asymptote at $y = 3$ (it never goes above 3 on my graph window)."

What is wrong with the student's reasoning?

A.

The student is correct — $\log_2(x)$ does have a horizontal asymptote at $y = 3$ .

B.

The logarithmic function only has a vertical asymptote, not a horizontal one. The graph grows without bound, just very slowly.

C.

The student should have used a larger graph window — the asymptote is at $y = 100$ .

D.

The student confused the horizontal and vertical asymptotes — the asymptote is at $x = 0$ , not a horizontal one.

F

Challenge / Extension

These problems extend your thinking and connect to future topics.

1.

The function $f(x) = 5^x$ and the function $g(x) = \log_5(x)$ are inverse functions.
Explain, using the concept of swapping inputs and outputs, why the graph of $g$ is
the reflection of the graph of $f$ over the line $y = x$ .
Include at least one specific coordinate pair from each graph in your explanation.

2.

A sine function has amplitude 6, period $4\pi$ , and midline $y = -3$ .
The function is of the form $y = A\sin(Bx) + D$ with $A > 0$ and $B > 0$ .