Q. the graph of (y = 5x^2) is the graph of (y = x^2).

Problem

Compare the graph of \(y = 5x^2\) with the graph of \(y = x^2\). Explain step by step what the relationship is and how to produce the graph of \(y = 5x^2\) from the graph of \(y = x^2\).

1. Write each function clearly.

   Base parabola: \(y = x^2\).

   Transformed parabola: \(y = 5x^2\), which can be written as \(y = 5\,(x^2) = 5\cdot f(x)\) where \(f(x)=x^2\).

2. Interpret the algebraic change.

   Multiplying \(f(x)\) by 5 means every output (every y-value) of the base graph is multiplied by 5. For each x, the new y is

   \(y_{\text{new}} = 5\cdot y_{\text{old}}\).

   So the mapping of points is \((x, y_{\text{old}})\) becomes \((x, 5\,y_{\text{old}})\).

3. Describe the geometric transformation.

   This multiplication of y-values is a vertical stretch by a factor of 5 (a vertical dilation with scale factor 5) about the x-axis. Distances from the x-axis are multiplied by 5; x-coordinates do not change.

4. Compare key features.
   • Vertex: both parabolas have the same vertex at \((0,0)\) because \(5\cdot 0^2 = 0\).
   • Axis of symmetry: both have the y-axis as axis of symmetry (the line \(x=0\)).
   • Direction and concavity: both open upward (concave up) because the coefficient of \(x^2\) is positive in both cases.
   • Steepness/narrowness: \(y = 5x^2\) is steeper (narrower) than \(y = x^2\) because y-values grow 5 times faster for the same x.
   • Intercepts: solving \(5x^2=0\) gives \(x=0\), so the only x-intercept and the y-intercept are both at the origin \((0,0)\).
5. Compare specific points (numerical check).

   Take sample x-values and compute both y-values:

   For \(x=-2\): \(y_{base}=(-2)^2=4\). For transformed: \(y_{new}=5\cdot 4=20\). Point moves from \((-2,4)\) to \((-2,20)\).

   For \(x=-1\): \(y_{base}=1\). New: \(y_{new}=5\). Point moves from \((-1,1)\) to \((-1,5)\).

   For \(x=0\): \(y_{base}=0\). New: \(y_{new}=0\). Point stays \((0,0)\).

   For \(x=1\): \(y_{base}=1\). New: \(y_{new}=5\). Point moves from \((1,1)\) to \((1,5)\).

   For \(x=2\): \(y_{base}=4\). New: \(y_{new}=20\). Point moves from \((2,4)\) to \((2,20)\).

6. How to draw \(y = 5x^2\) from \(y = x^2\) — step-by-step instructions.
   1. Plot several points of the base graph \(y = x^2\), for example \((-2,4)\), \((-1,1)\), \((0,0)\), \((1,1)\), \((2,4)\).
   2. Multiply each y-coordinate by 5 to get the new points: \((-2,20)\), \((-1,5)\), \((0,0)\), \((1,5)\), \((2,20)\).
   3. Plot these new points on the same axes.
   4. Draw a smooth parabola through the new points; it will be narrower and steeper than \(y = x^2\) but share the same vertex and axis.
7. Conclusion (concise answer).

   The graph of \(y = 5x^2\) is not the same as the graph of \(y = x^2\). It is the graph of \(y = x^2\) vertically stretched by a factor of 5 about the x-axis. Every y-value of \(y = x^2\) is multiplied by 5 to obtain \(y = 5x^2\).