8.12 Comparing exponential and quadratic functions
Exponential functions have the form $A\left(a^x\right)+k$A(ax)+k. Quadratic functions have the form $ax^2+bx+c$ax2+bx+c. Here are some features of both functions.
| Exponential functions | Quadratic functions | |
|---|---|---|
| $x$x-intercepts | Either zero or one $x$x-intercept | Either zero, one or two $x$x-intercepts |
| $y$y-intercept | One $y$y-intercept | One $y$y-intercept |
| Asymptotes | One horizontal asymptote | No asymptotes |
| Turning points | No turning points | One turning point |
| Direction | Either always increasing or always decreasing | Increasing on one side of the turning point and decreasing on the other. |
Practice questions
Question 1
Consider the functions $f\left(x\right)=9x^2$f(x)=9x2 and $g\left(x\right)=9^x$g(x)=9x.
Complete the table of values.
$x$x $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $f\left(x\right)$f(x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $g\left(x\right)$g(x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Select the graph for $f\left(x\right)$f(x).
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DSelect the graph for $g\left(x\right)$g(x).
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DWhich function has a minimum value?
$f\left(x\right)$f(x)
A$g\left(x\right)$g(x)
BBoth
CNeither
DWhich function has a horizontal asymptote?
$f\left(x\right)$f(x)
A$g\left(x\right)$g(x)
BBoth
CNeither
D
Question 2
Consider the functions $f\left(x\right)=2x^2-6$f(x)=2x2−6 and $g\left(x\right)=2^x-6$g(x)=2x−6.
Complete the table of values.
$x$x $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $f\left(x\right)$f(x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $g\left(x\right)$g(x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Select the graph for $f\left(x\right)$f(x).
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DSelect the graph for $g\left(x\right)$g(x).
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DWhich function is increasing when $x<0$x<0?
$f\left(x\right)$f(x)
A$g\left(x\right)$g(x)
BBoth
CNeither
DWhich function is increasing when $x>0$x>0?
$f\left(x\right)$f(x)
A$g\left(x\right)$g(x)
BBoth
CNeither
D
Question 3
Consider the functions $f\left(x\right)=x^2+2x+5$f(x)=x2+2x+5 and $g\left(x\right)=6^x-8$g(x)=6x−8.
Complete the table of values.
$x$x $-2$−2 $-1$−1 $0$0 $1$1 $2$2 $f\left(x\right)$f(x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $g\left(x\right)$g(x) $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ $\editable{}$ Select the graph for $f\left(x\right)$f(x).
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DSelect the graph for $g\left(x\right)$g(x).
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DWhich function has a minimum value?
$f\left(x\right)$f(x)
A$g\left(x\right)$g(x)
BBoth
CNeither
DWhich function has a horizontal asymptote?
$f\left(x\right)$f(x)
A$g\left(x\right)$g(x)
BBoth
CNeither
D