Question

12 consider the graph of the equation $y = 2^x$. what can we say about the $y$-value of every point on the graph? the $y$-value of every point on the graph is an integer. the $y$-value of most points on the graph is positive, and the $y$-value at one point is 0. the $y$-value of most points of the graph is greater than 1. the $y$-value of every point on the graph is positive.

Explanation:

Step1: Analyze the function \( y = 2^x \)

The function \( y = 2^x \) is an exponential function with base \( 2>1 \). For any real number \( x \), the value of \( 2^x \) is always positive. When \( x = 0 \), \( y=2^0 = 1 \). When \( x>0 \), \( y = 2^x>1 \) (since the function is increasing). When \( x<0 \), \( 01 \) when \( |x|>0 \)).

Step2: Evaluate each option

• Option 1: "The \( y \)-value of every point on the graph is an integer."

For example, when \( x = 0.5 \), \( y = 2^{0.5}=\sqrt{2}\approx1.414 \), which is not an integer. So this option is incorrect.

• Option 2: "The \( y \)-value of most points on the graph is positive, and the \( y \)-value at one point is 0."

But for \( y = 2^x \), \( 2^x>0 \) for all real \( x \), so \( y \) is never 0. This option is incorrect.

• Option 3: "The \( y \)-value of most points of the graph is greater than 1."

When \( x<0 \), \( 0

• Option 4: "The \( y \)-value of every point on the graph is positive."

Since \( 2^x>0 \) for all real \( x \) (exponential function with positive base), this statement is true.

Answer:

The \( y \)-value of every point on the graph is positive.