6.2 Graphs of Exponential Functions

Solve $42 = 1.2 \left(5\right)^{x} + 2.8$ graphically. Round to the nearest thousandth.

Solve $4 = 7.85 \left(1.15\right)^{x} - 2.27$ graphically. Round to the nearest thousandth.

Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.

• $f \left(x\right) = e^{x}$ is vertically stretched by a factor of $2$ , reflected across the y-axis, and then shifted up $4$ units.

Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.

• $f \left(x\right) = e^{x}$ is compressed vertically by a factor of $\frac{1}{3} ,$ reflected across the x-axis and then shifted down $2$ units.

What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?

The graph of $f \left(x\right) = 3^{x}$ is reflected about the y-axis and stretched vertically by a factor of $4.$ What is the equation of the new function, $g \left(x\right) ?$ State its y-intercept, domain, and range.

The graph of $f \left(x\right) = 10^{x}$ is reflected about the x-axis and shifted upward $7$ units. What is the equation of the new function, $g \left(x\right) ?$ State its y-intercept, domain, and range.

The graph of $f \left(x\right) = - \frac{1}{2} \left(\frac{1}{4}\right)^{x - 2} + 4$ is shifted downward $4$ units, and then shifted left $2$ units, stretched vertically by a factor of $4 ,$ and reflected about the x-axis. What is the equation of the new function, $g \left(x\right) ?$ State its y-intercept, domain, and range.

$f \left(x\right) = 2 \left(0.69\right)^{x}$

$f \left(x\right) = 2 \left(0.81\right)^{x}$

$f \left(x\right) = 2 \left(1.59\right)^{x}$

Which graph has the largest value for $b ?$

Which graph has the largest value for $a ?$

Describe the end behavior of the graphs of the functions.

$f \left(x\right) = - 5 \left(4\right)^{x} - 1$

Describe the end behavior of the graphs of the functions.

$f \left(x\right) = 3 \left(4\right)^{- x} + 2$

Start with the graph of $f \left(x\right) = 4^{x} .$ Then write a function that results from the given transformation.

Shift $f \left(x\right)$ 3 units downward

Start with the graph of $f \left(x\right) = 4^{x} .$ Then write a function that results from the given transformation.

Shift $f \left(x\right)$ 5 units right

Start with the graph of $f \left(x\right) = 4^{x} .$ Then write a function that results from the given transformation.

Reflect $f \left(x\right)$ about the y-axis

Evaluate the exponential functions for the indicated value of $x .$

$g \left(x\right) = \frac{1}{3} \left(7\right)^{x - 2}$ for $g \left(6\right) .$

Evaluate the exponential functions for the indicated value of $x .$

$h \left(x\right) = - \frac{1}{2} \left(\frac{1}{2}\right)^{x} + 6$ for $h \left(- 7\right) .$

Use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.

$116 = \frac{1}{4} \left(\frac{1}{8}\right)^{x}$

Use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.

$5 = 3 \left(\frac{1}{2}\right)^{x - 1} - 2$

Explore and discuss the graphs of $F \left(x\right) = \left(b\right)^{x}$ and $G \left(x\right) = \left(\frac{1}{b}\right)^{x} .$ Then make a conjecture about the relationship between the graphs of the functions $b^{x}$ and $\left(\frac{1}{b}\right)^{x}$ for any real number $b > 0.$

Explore and discuss the graphs of $f \left(x\right) = 4^{x} ,$ $g \left(x\right) = 4^{x - 2} ,$ and $h \left(x\right) = \left(\frac{1}{16}\right) 4^{x} .$ Then make a conjecture about the relationship between the graphs of the functions $b^{x}$ and $\left(\frac{1}{b^{n}}\right) b^{x}$ for any real number n and real number $b > 0.$