Draw the graphs of the equations below according to the Graph Papers indicated: First Graph Paper Original graph \( (M): y=x^{2} \) Graph 1: \( y=2 x^{2} \) Graph 2: \( y=\frac{1}{2} x^{2} \) Graph 3: \( y=-2 x^{2} \)

### Graphing the Given Equations We are given the base parabola \[ y = x^{2}, \] and three related graphs: - **Graph 1:** \(\ y = 2x^{2}\) - **Graph 2:** \(\ y = \frac{1}{2}x^{2}\) - **Graph 3:** \(\ y = -2x^{2}\) Below are the step-by-step analyses and instructions for drawing each graph on the same coordinate grid. --- #### 1. Base Graph: \(\boldsymbol{y=x^2}\) - **Shape:** Parabola opening upward. - **Vertex:** \((0,0)\). - **Symmetry:** About the \(y\)-axis. - **Key Points:** - \((1,1)\), \((-1,1)\), - \((2,4)\), \((-2,4)\). **Drawing Tip:** Plot the vertex at the origin and mark symmetric points on either side of the \(y\)-axis. --- #### 2. Graph 1: \(\boldsymbol{y=2x^2}\) - **Transformation:** Vertical stretch by a factor of 2. - **Effect:** For each \(x\), the \(y\) value is doubled compared to \(y=x^2\). This makes the graph **narrower**. - **Key Points:** - At \(x=1\): \(y=2 \times 1^2 = 2\). - At \(x=2\): \(y=2 \times 2^2 = 8\). - At \(x=-1\) and \(x=-2\) the points mirror these values. **Drawing Tip:** Use the same vertex at \((0,0)\) and plot the new points. The parabola will be steeper than \(y=x^2\). --- #### 3. Graph 2: \(\boldsymbol{y=\frac{1}{2}x^2}\) - **Transformation:** Vertical shrink (or compression) by a factor of \(\frac{1}{2}\). - **Effect:** For each \(x\), the \(y\)-value is half of that in \(y=x^2\). This makes the graph **wider**. - **Key Points:** - At \(x=1\): \(y=\frac{1}{2} \times 1^2 = \frac{1}{2}\). - At \(x=2\): \(y=\frac{1}{2} \times 2^2 = 2\). - Again, symmetric points appear for negative \(x\). **Drawing Tip:** Start at the vertex \((0,0)\) and plot the new points. Notice the parabola spreads out more compared to \(y=x^2\). --- #### 4. Graph 3: \(\boldsymbol{y=-2x^2}\) - **Transformation:** Vertical reflection across the \(x\)-axis combined with a vertical stretch by a factor of 2. - **Effect:** The original upward opening parabola is flipped to open downward. - **Key Points:** - At \(x=1\): \(y=-2 \times 1^2 = -2\). - At \(x=2\): \(y=-2 \times 2^2 = -8\). - The graph remains symmetric about the \(y\)-axis with the vertex at \((0,0)\). **Drawing Tip:** Plot the vertex at \((0,0)\). Instead of rising, the branches of the parabola will extend downward, becoming steeper due to the factor of 2. --- ### Summary of Graph Features on the Same Graph Paper - **Vertex:** All graphs share the same vertex at \((0,0)\). - **Symmetry:** All parabolas are symmetric about the \(y\)-axis. - **Comparison:** - \(y = 2x^2\) is narrower than the base graph. - \(y = \frac{1}{2}x^2\) is wider than the base graph. - \(y = -2x^2\) opens downward with a narrow shape, mirroring the steepness of \(y = 2x^2\) but in the negative \(y\)-direction. --- ### Final Drawing Instructions 1. **Draw the Coordinate Axes:** Ensure that both positive and negative parts of the \(x\)- and \(y\)-axes are clearly marked. 2. **Plot the Vertex:** Mark the common vertex \((0,0)\) for all graphs. 3. **Plot Key Points for Each Graph:** - For \(y = x^2\): Plot \((1,1)\), \((-1,1)\), \((2,4)\), \((-2,4)\). - For \(y = 2x^2\): Plot \((1,2)\), \((-1,2)\), \((2,8)\), \((-2,8)\). - For \(y = \frac{1}{2}x^2\): Plot \((1,\frac{1}{2})\), \((-1,\frac{1}{2})\), \((2,2)\), \((-2,2)\). - For \(y = -2x^2\): Plot \((1,-2)\), \((-1,-2)\), \((2,-8)\), \((-2,-8)\). 4. **Sketch the Parabolas:** Connect the points smoothly for each equation respecting the symmetry about the \(y\)-axis. 5. **Differentiate the Graphs:** Use distinct colors or line styles for each graph if drawing on the same paper. Following these steps will give you a clear picture of how the scaling (vertical stretch and compression) and reflection affect the shape and orientation of the parabolas.