graph: $f(x)=2(2)^{x}$\nwhich statements are true about the graph? check all that apply.\nthe $y$-intercept…

graph: $f(x)=2(2)^{x}$\nwhich statements are true about the graph? check all that apply.\nthe $y$-intercept is $(0,2)$.\nthe $x$-intercept is $(4,0)$.\n$f(1)=2$\n$f(2)=8$\nit is a shrink of an exponential growth function.

graph: $f(x)=2(2)^{x}$\nwhich statements are true about the graph? check all that apply.\nthe $y$-intercept is $(0,2)$.\nthe $x$-intercept is $(4,0)$.\n$f(1)=2$\n$f(2)=8$\nit is a shrink of an exponential growth function.

Answer

Explanation:

Step1: Find the y - intercept

For the y - intercept, set (x = 0). Then (f(0)=2(2)^{0}). Since (a^{0}=1) ((a\neq0)), (f(0)=2\times1 = 2). So the y - intercept is ((0,2)).

Step2: Check for the x - intercept

Set (y = 0), then (0=2(2)^{x}). The equation (2(2)^{x}=0) has no solution because (2^{x}>0) for all real (x) and (2>0), so there is no x - intercept.

Step3: Calculate (f(1))

Substitute (x = 1) into (f(x)=2(2)^{x}), we get (f(1)=2(2)^{1}). Using the rule (a^{m}\times a^{n}=a^{m + n}) (here (a = 2,m = 1,n=1)), (f(1)=2\times2=4\neq2).

Step4: Calculate (f(2))

Substitute (x = 2) into (f(x)=2(2)^{x}), we get (f(2)=2(2)^{2}). First, (2^{2}=4), then (2\times4 = 8).

Step5: Analyze the transformation

The parent function is (y = 2^{x}). The function (y=2(2)^{x}) is a vertical stretch (not a shrink) of (y = 2^{x}) by a factor of 2.

Answer:

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