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ACT Math : How to graph an exponential function

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Example Questions

Example Question #2 : Graph A Polynomial Function

Give the -intercept of the graph of the function

$f(x) = 5 \cdot 4^{x- 3}- 3$

Round to the nearest tenth, if applicable.

Possible Answers:

( -2.92, 0)

(2.63,0)

(3.65,0)

( -3.08, 0)

The graph has no -interceptx

Correct answer:

(2.63,0)

Explanation:

The -intercept is (a,0) , where f(a)= 0 :

$f(x) = 5 \cdot 4^{x- 3}- 3$

$5 \cdot 4^{a- 3}- 3 = 0$

$5 \cdot 4^{a- 3}- 3 + 3= 0 + 3$

$5 \cdot 4^{a- 3}= 3$

$5 \cdot 4^{a- 3} \div 5= 3 \div 5$

$4^{a- 3} = 0.6$

$\ln \left (4^{a- 3} \right )=\ln 0.6$

$(a- 3)\ln 4 =\ln 0.6$

$a- 3=\frac{\ln 0.6}{\ln 4 } \approx \frac{-0.5108}{1.3863}\approx -0.3685$

$a- 3+3 \approx -0.3685 + 3$

$a \approx 2.63$

The -intercept is (2.63,0) .

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Example Question #1 : How To Graph An Exponential Function

Give the -intercept of the graph of the function

$f(x) = 5 \cdot 4^{x- 3}- 3$

Round to the nearest hundredth, if applicable.

Possible Answers:

(0, -2.92)

(0, -3.08)

The graph has no -intercept

( 0, 2.63)

(0,3.65)

Correct answer:

(0, -2.92)

Explanation:

The -intercept is (0, f(0)) :

$f(x) = 5 \cdot 4^{x- 3}- 3$

$f(0) = 5 \cdot 4^{0- 3}- 3$

$f(0) = 5 \cdot 4^{ - 3}- 3$

$f(0) = 5 \cdot \frac{1}{64}- 3$

$f(0) = \frac{5}{64}- 3$

$f(0) \approx 0.08 - 3 \approx -2.92$

(0, -2.92) is the -intercept.

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Example Question #1 : How To Graph An Exponential Function

Give the vertical asymptote of the graph of the function

$g(x) = 16 \cdot 4^{x}- 3$

Possible Answers:

y = -2

The graph of has no vertical asymptote.

x = -3

x = 3

y = 2

Correct answer:

The graph of has no vertical asymptote.

Explanation:

Since 4 can be raised to the power of any real number, the domain of is the set of all real numbers. Therefore, there is no vertical asymptote of the graph of .

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Example Question #3 : How To Graph An Exponential Function

Give the horizontal asymptote of the graph of the function

$g(x) = 9 \cdot 3^{x}- 3$

Possible Answers:

y = 2

The graph has no horizontal asymptote.

y = 3

y = -3

y = -2

Correct answer:

y = -3

Explanation:

We can rewrite this as follows:

$g(x) = 9 \cdot 3^{x}- 3$

$g(x) = 3^{2} \cdot 3^{x}- 3$

$g(x) = 3^{x+2}- 3$

This is a translation of the graph of $f(x) = 3^{x}$ , which has y = 0 as its horizontal asymptote, to the right two units and down three units. Because of the latter translation, the horizontal asymptote is y = -3 .

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Example Question #1 : How To Graph An Exponential Function

If the functions

$f(x) = e^{x}+ 9$

$g(x)= 4e^{x}- 7$

were graphed on the same coordinate axes, what would be the -coordinate of their point of intersection?

Round to the nearest tenth, if applicable.

Possible Answers:

The graphs of and would not intersect.

2.7

14.3

1.7

5.3

Correct answer:

14.3

Explanation:

We can rewrite the statements using for both f(x) and g(x) as follows:

$y = e^{x}+ 9$

$y = 4e^{x}- 7$

To solve this, we can multiply the first equation by , then add:

$-4y = -4e^{x}-36$

$\underline{y = 4e^{x} \; \; - 7}$

-3y =

$-3y \div (-3) = -43 \div (-3 )$

$y \approx 14.3$

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Example Question #5 : How To Graph An Exponential Function

If the functions

$f(x) = e^{x}+ 9$

$g(x) = 4e^{x}- 7$

were graphed on the same coordinate axes, what would be the -coordinate of their point of intersection?

Round to the nearest tenth, if applicable.

Possible Answers:

The graphs of and would not intersect.

1.7

14.3

2.7

5.3

Correct answer:

1.7

Explanation:

We can rewrite the statements using for both f(x) and g(x) as follows:

$y = e^{x}+ 9$

$y = 4e^{x}- 7$

To solve this, we can set the expressions equal, as follows:

$4e^{x}- 7 = e^{x}+ 9$

$4e^{x}- 7 - e^{x} + 7 = e^{x}+ 9 - e^{x} + 7$

$3e^{x} = 16$

$3e^{x}\div 3 = 16 \div 3$

$e^{x} = \approc 5.3333$

$x \approx \ln 5.333 \approx 1.7$

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ACT Math : How to graph an exponential function

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #2 : Graph A Polynomial Function

Example Question #1 : How To Graph An Exponential Function

Example Question #1 : How To Graph An Exponential Function

Example Question #3 : How To Graph An Exponential Function

Example Question #1 : How To Graph An Exponential Function

Example Question #5 : How To Graph An Exponential Function

All ACT Math Resources

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