Problem

Compare \(f(x)=(x-2)^{2}+1\) and \(g(x)=3(x-2)^{2}+1\). List \(a\), \(h\), and \(k\) for each, identify all shared graph features, and explain quantitatively how \(g\)'s vertical displacement from \(y=1\) and width differ from \(f\)'s.

Big Picture

What this problem is really about

Comparing vertex-form parameters one role at a time separates shared location from different scale. We’ll match h and k to identify the common vertex and axis, compare the signs for opening, and relate the two a-values through vertical displacement from the shared vertex level to explain the width difference.

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Both have vertex (−2, 1) and axis x = −2, and g is narrower. The expression x − 2 shifts the vertex to x = 2, not x = −2.

Both are upward-opening quadratics with vertex (2, 1) and axis x = 2. The factor 3 makes g vertically steeper, so g is narrower than f.

Both use the same parameters, but only g has a vertical scale factor. This does not identify their common vertex, axis, opening, or the graph effect.

Both have vertex (2, 1), but g is horizontally compressed by 3. The 3 multiplies the output, so its effect is vertical rather than horizontal.

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Compare \(f(x)=(x-2)^{2}+1\) and \(g(x)=3(x-2)^{2}+1\). List \(a\), \(h\), and \(k\) for each, identify all shared graph features, and explain quantitatively how \(g\)'s vertical displacement from \(y=1\) and width differ from \(f\)'s.

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