1. The problem states that the black graph is $y=f(x)$ passing through $(1,1)$ and the red graph passes through $(1,4)$ and is narrower. 2. We are asked to choose the equation for the red graph from these options: $y=4f(x)$, $y=f(x/4)$, $y=\frac{f(x)}{4}$, $y=f(4x)$. 3. The "big idea of opposites" refers to how transformations affect the graph. 4. Multiplying $f(x)$ by 4 as in $y=4f(x)$ vertically stretches the graph, making it taller and passing through $(1,4)$ if $f(1)=1$. 5. Replacing $x$ by $x/4$ as in $y=f(x/4)$ horizontally stretches the graph, making it wider, which contradicts the red graph being narrower. 6. Dividing $f(x)$ by 4 as in $y=\frac{f(x)}{4}$ vertically compresses the graph, making it shorter, so it cannot pass through $(1,4)$. 7. Replacing $x$ by $4x$ as in $y=f(4x)$ horizontally compresses the graph, making it narrower, but the point $(1,4)$ would not be on this graph if $f(1)=1$. 8. Since the red graph passes through $(1,4)$ and is narrower, the vertical stretch $y=4f(x)$ fits both conditions. Final answer: $$y=4f(x)$$