Question

the graphs of the functions \\(f\\) and \\(g\\) are given. which of the following defines \\(g\\) in terms of \\(f\\)?

Explanation:

Identify key points on the graph of f

We examine the graph of \(f\) to find clear coordinate points \((x, f(x))\):

• The \(y\)-intercept is at \((0, 2)\), so \(f(0) = 2\).
• The \(x\)-intercept on the left is at \((-2, 0)\), so \(f(-2) = 0\).
• The \(x\)-intercept on the right is at \((2, 0)\), so \(f(2) = 0\).
• Another point on the right is at \((1, 0.5)\) or similar, but the intercepts are the most reliable.

Identify corresponding points on the graph of g

We examine the graph of \(g\) to find corresponding features and coordinate points \((x, g(x))\):

• The \(y\)-intercept is at \((0, 5)\), so \(g(0) = 5\).
• The left \(x\)-intercept is at \((-0.5, 0)\), so \(g(-0.5) = 0\).
• The right \(x\)-intercept is at \((0.5, 0)\), so \(g(0.5) = 0\).

Analyze horizontal transformation

Let's compare the \(x\)-intercepts of \(f\) and \(g\):

• For \(f(x)\), the intercepts are at \(x = -2\) and \(x = 2\).
• For \(g(x)\), the intercepts are at \(x = -0.5\) and \(x = 0.5\).
• The horizontal distance between the intercepts has been compressed by a factor of \(4\) (from a width of \(4\) to a width of \(1\)).
• A horizontal compression by a factor of \(4\) is represented by replacing the input \(x\) with \(4x\). Thus, the transformation involves \(f(4x)\).

Analyze vertical transformation

Let's check the value of \(f(4x)\) at \(x = 0\):

• \(f(4 \cdot 0) = f(0) = 2\).
• However, the observed \(y\)-intercept of \(g(x)\) is \(g(0) = 5\).
• To scale the output from \(2\) to \(5\), we must multiply the function by a vertical stretch factor \(a\):

$$ a \cdot 2 = 5 \implies a = 2.5 = \frac{5}{2} $$

• Therefore, the vertical scaling factor is \(\frac{5}{2}\) (or \(2.5\)).

Combine transformations to define g(x)

Combining the horizontal compression and vertical stretch:

• The horizontal compression gives the inner term \(f(4x)\).
• The vertical stretch gives the outer multiplier \(\frac{5}{2}\).
• Thus, the function \(g(x)\) in terms of \(f(x)\) is:

$$ g(x) = 2.5f(4x) \quad \text{or} \quad g(x) = \frac{5}{2}f(4x) $$

Answer:

$$g(x) = 2.5f(4x)$$