Question

graph: $f(x) = 2(2)^x$
which statements are true about the graph? choose two correct answers.
the $x$-intercept is $(4, 0)$.
the $y$-intercept is $(0, 2)$.
$f(2) = 8$
it is a shrink of an exponential growth function.
$f(1) = 2$
click or tap the graph to plot a point.

Explanation:

Step1: Check x - intercept

To find the x - intercept, set \(y = f(x)=0\). So, \(0 = 2(2)^{x}\). But \(2(2)^{x}>0\) for all real \(x\) (since exponential function \(a^{x}\) with \(a > 0,a
eq1\) is always positive and multiplied by 2). So, there is no x - intercept. So, the statement "The x - intercept is \((4,0)\)" is false.

Step2: Check y - intercept

The y - intercept is found by setting \(x = 0\). Substitute \(x = 0\) into \(f(x)=2(2)^{x}\). Then \(f(0)=2(2)^{0}\). Since \(a^{0}=1\) for \(a
eq0\), \(2^{0}=1\). So, \(f(0)=2\times1 = 2\). So, the y - intercept is \((0,2)\). This statement is true.

Step3: Check \(f(2)\)

Substitute \(x = 2\) into \(f(x)=2(2)^{x}\). Then \(f(2)=2(2)^{2}\). \(2^{2}=4\), so \(f(2)=2\times4 = 8\). So, the statement \(f(2)=8\) is true.

Step4: Check exponential shrink/growth

The general form of an exponential function is \(y = a(b)^{x}\). If \(b>1\), it is a growth function. Here, \(b = 2>1\), so it is a growth function. The coefficient \(a = 2>1\), so it is a vertical stretch (not shrink) of the parent function \(y = 2^{x}\). So, the statement "It is a shrink of an exponential growth function" is false.

Step5: Check \(f(1)\)

Substitute \(x = 1\) into \(f(x)=2(2)^{x}\). Then \(f(1)=2(2)^{1}=2\times2 = 4
eq2\). So, the statement \(f(1)=2\) is false.

Answer:

The two correct statements are:

• The \(y\) - intercept is \((0,2)\)
• \(f(2)=8\)