How to graph an exponential function

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ACT Math › How to graph an exponential function

Questions 1 - 6

1

Give the -intercept of the graph of the function

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

Round to the nearest tenth, if applicable.

The graph has no -interceptx

Explanation

The -intercept is , where :

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

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

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

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

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

$4^{a- 3} = 0.6$

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

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

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

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

$a \approx 2.63$

The -intercept is .

2

Give the -intercept of the graph of the function

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

Round to the nearest hundredth, if applicable.

The graph has no -intercept

Explanation

The -intercept is :

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

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

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

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

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

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

is the -intercept.

3

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.

The graphs of and would not intersect.

Explanation

We can rewrite the statements using for both and as follows:

$y = e^{x}+ 9$

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

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

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

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

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

$y \approx 14.3$

4

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.

The graphs of and would not intersect.

Explanation

We can rewrite the statements using for both and as follows:

$y = e^{x}+ 9$

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

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

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

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

$3e^{x} = 16$

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

$e^{x} = \approc 5.3333$

$x \approx \ln 5.333 \approx 1.7$

5

Give the horizontal asymptote of the graph of the function

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

The graph has no horizontal asymptote.

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 as its horizontal asymptote, to the right two units and down three units. Because of the latter translation, the horizontal asymptote is .

6

Give the vertical asymptote of the graph of the function

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

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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