Question

question 27
match each equation with a graph above.
$y = - 2 ^ { x }$ a. blue (b)
$y = 2 ^ { x }$ b. orange (o)
$y = 2 ^ { - x }$ c. red (r)
$y = - 2 ^ { - x }$ d. green (g)

Explanation:

Step1: Analyze \(y = 2^{x}\)

The function \(y = 2^{x}\) is an exponential growth function. When \(x = 0\), \(y=2^{0}=1\). As \(x\) increases, \(y\) increases rapidly. This matches the orange curve (since exponential growth curves go up as \(x\) increases), so \(y = 2^{x}\) corresponds to b. orange (O).

Step2: Analyze \(y = 2^{-x}=( \frac{1}{2})^{x}\)

The function \(y = 2^{-x}\) is an exponential decay function. When \(x = 0\), \(y = 2^{-0}=1\). As \(x\) increases, \(y\) decreases. This matches the red curve (exponential decay curves go down as \(x\) increases in a non - negative direction), so \(y = 2^{-x}\) corresponds to c. red (R).

Step3: Analyze \(y=-2^{x}\)

The function \(y =-2^{x}\) is the reflection of \(y = 2^{x}\) over the \(x\) - axis. When \(x = 0\), \(y=-2^{0}=-1\). As \(x\) increases, \(y\) becomes more negative. This matches the blue curve (since it has a negative \(y\) - value trend similar to the reflection of an exponential growth), so \(y=-2^{x}\) corresponds to a. blue (B).

Step4: Analyze \(y=-2^{-x}\)

The function \(y=-2^{-x}\) is the reflection of \(y = 2^{-x}\) over the \(x\) - axis. When \(x = 0\), \(y=-2^{-0}=-1\). As \(x\) increases, \(y\) approaches \(0\) from the negative side. This matches the green curve (reflection of an exponential decay over the \(x\) - axis), so \(y=-2^{-x}\) corresponds to d. green (G).

Answer:

\(y=-2^{x}\): a. blue (B)

\(y = 2^{x}\): b. orange (O)

\(y = 2^{-x}\): c. red (R)

\(y=-2^{-x}\): d. green (G)