6.2 Graphs of exponential functions  (Page 6/6)

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

What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?

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

The graph of $f(x)={\left(\frac{1}{2}\right)}^{-x}$ is reflected about the y -axis and compressed vertically by a factor of $\frac{1}{5}.$ What is the equation of the new function, $g(x)?$ State its y -intercept, domain, and range.

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

The graph of $f(x)={\left(1.68\right)}^x$ is shifted right $3$ units, stretched vertically by a factor of $2,$ reflected about the x -axis, and then shifted downward $3$ units. What is the equation of the new function, $g(x)?$ State its y -intercept (to the nearest thousandth), domain, and range.

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

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

$g(x)=-2{\left(0.25\right)}^x$

$h(x)=6{\left(1.75\right)}^{-x}$

$f(x)=3{\left(\frac{1}{4}\right)}^x,$ $g(x)=3{\left(2\right)}^x,$ and $h(x)=3{\left(4\right)}^x$

$f(x)=\frac{1}{4}{\left(3\right)}^x,$ $g(x)=2{\left(3\right)}^x,$ and $h(x)=4{\left(3\right)}^x$

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

$f\left(x\right)=2{\left(1.28\right)}^x$

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

$f\left(x\right)=4{\left(1.28\right)}^x$

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

$f\left(x\right)=4{\left(0.69\right)}^x$

Which graph has the largest value for $b?$

Which graph has the smallest value for $b?$

Which graph has the largest value for $a?$

Which graph has the smallest value for $a?$

$f(x)=\frac{1}{2}{\left(4\right)}^x$

$f(x)=3{\left(0.75\right)}^x-1$

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

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

$h\left(x\right)=2^x+3$

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

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

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

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

Shift $f(x)$ 4 units upward

Shift $f(x)$ 3 units downward

Shift $f(x)$ 2 units left

Shift $f(x)$ 5 units right

Reflect $f(x)$ about the x -axis

Reflect $f(x)$ about the y -axis

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

$f(x)=4{(2)}^{x-1}-2$ for $f(5).$

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

$-50=-{\left(\frac{1}{2}\right)}^{-x}$

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

$12=2{\left(3\right)}^x+1$

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

$-30=-4{\left(2\right)}^{x+2}+2$

Explore and discuss the graphs of $F(x)={\left(b\right)}^x$ and $G(x)={\left(\frac{1}{b}\right)}^x.$ 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 $b>0.$

Prove the conjecture made in the previous exercise.

Explore and discuss the graphs of $f(x)=4^x,$ $g(x)=4^{x-2},$ and $h(x)=\left(\frac{1}{16}\right)4^x.$ 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 $b>0.$

Prove the conjecture made in the previous exercise.