Sketching the Parabola Exercise 1: \( \begin{array}{lll}\text { 1) Sketch the graphs of the } f_{0} \\ \text { (a) } y=2(x-3)^{2}-2 & \text { c) } h(x)=-(x-3)^{2}+5 & \text { e) } m(x)=-(x+2)^{2}-1 \\ \text { /b) } f(x)=-(x+2)^{2}+4 & \text { d) } p(x)=1 / 2(x+2)^{2}+2 & \text { f) } f: x \rightarrow(x+3)^{2}-4\end{array} \)

Certainly! Let's analyze and sketch each of the given quadratic functions. Quadratic functions are typically expressed in the **vertex form**: \[ y = a(x - h)^2 + k \] where: - \((h, k)\) is the **vertex** of the parabola. - \(a\) determines the **direction** and **width** of the parabola: - If \(a > 0\), the parabola **opens upwards**. - If \(a < 0\), the parabola **opens downwards**. - The larger \(|a|\) is, the **narrower** the parabola. - The smaller \(|a|\) is (but \(a \neq 0\)), the **wider** the parabola. Let's go through each function step-by-step: --- ### **a) \( y = 2(x - 3)^2 - 2 \)** - **Vertex Form Analysis:** - \( a = 2 \) (positive, so parabola opens **upwards**) - \( h = 3 \) - \( k = -2 \) - **Vertex:** \( (3, -2) \) - **Axis of Symmetry:** \( x = 3 \) - **Width:** Since \(|a| = 2 > 1\), the parabola is **narrower** than the standard \( y = x^2 \). - **Sketching Steps:** 1. Plot the vertex at \( (3, -2) \). 2. Draw the axis of symmetry \( x = 3 \). 3. Since \(a = 2\), choose points on either side of the vertex (e.g., \( x = 2 \) and \( x = 4 \)) to find corresponding \( y \)-values. 4. Plot these points and sketch the upward-opening parabola. --- ### **b) \( f(x) = -(x + 2)^2 + 4 \)** - **Vertex Form Analysis:** - \( a = -1 \) (negative, so parabola opens **downwards**) - \( h = -2 \) - \( k = 4 \) - **Vertex:** \( (-2, 4) \) - **Axis of Symmetry:** \( x = -2 \) - **Width:** Since \(|a| = 1\), the parabola has the **standard width**. - **Sketching Steps:** 1. Plot the vertex at \( (-2, 4) \). 2. Draw the axis of symmetry \( x = -2 \). 3. Choose points on either side of the vertex (e.g., \( x = -3 \) and \( x = -1 \)) to find corresponding \( y \)-values. 4. Plot these points and sketch the downward-opening parabola. --- ### **c) \( h(x) = -(x - 3)^2 + 5 \)** - **Vertex Form Analysis:** - \( a = -1 \) (negative, so parabola opens **downwards**) - \( h = 3 \) - \( k = 5 \) - **Vertex:** \( (3, 5) \) - **Axis of Symmetry:** \( x = 3 \) - **Width:** Standard width since \(|a| = 1\). - **Sketching Steps:** 1. Plot the vertex at \( (3, 5) \). 2. Draw the axis of symmetry \( x = 3 \). 3. Select points around the vertex (e.g., \( x = 2 \) and \( x = 4 \)) to determine \( y \)-values. 4. Plot these points and draw the downward-opening parabola. --- ### **d) \( p(x) = \frac{1}{2}(x + 2)^2 + 2 \)** - **Vertex Form Analysis:** - \( a = \frac{1}{2} \) (positive, so parabola opens **upwards**) - \( h = -2 \) - \( k = 2 \) - **Vertex:** \( (-2, 2) \) - **Axis of Symmetry:** \( x = -2 \) - **Width:** Since \(|a| = \frac{1}{2} < 1\), the parabola is **wider** than the standard \( y = x^2 \). - **Sketching Steps:** 1. Plot the vertex at \( (-2, 2) \). 2. Draw the axis of symmetry \( x = -2 \). 3. Choose points on either side (e.g., \( x = -3 \) and \( x = -1 \)) and calculate \( y \)-values. 4. Plot these points and sketch the upward-opening, wider parabola. --- ### **e) \( m(x) = -(x + 2)^2 - 1 \)** - **Vertex Form Analysis:** - \( a = -1 \) (negative, so parabola opens **downwards**) - \( h = -2 \) - \( k = -1 \) - **Vertex:** \( (-2, -1) \) - **Axis of Symmetry:** \( x = -2 \) - **Width:** Standard width since \(|a| = 1\). - **Sketching Steps:** 1. Plot the vertex at \( (-2, -1) \). 2. Draw the axis of symmetry \( x = -2 \). 3. Select points around the vertex (e.g., \( x = -3 \) and \( x = -1 \)) to find \( y \)-values. 4. Plot these points and draw the downward-opening parabola. --- ### **f) \( f(x) = (x + 3)^2 - 4 \)** - **Vertex Form Analysis:** - \( a = 1 \) (positive, so parabola opens **upwards**) - \( h = -3 \) - \( k = -4 \) - **Vertex:** \( (-3, -4) \) - **Axis of Symmetry:** \( x = -3 \) - **Width:** Standard width since \(|a| = 1\). - **Sketching Steps:** 1. Plot the vertex at \( (-3, -4) \). 2. Draw the axis of symmetry \( x = -3 \). 3. Choose points on either side of the vertex (e.g., \( x = -4 \) and \( x = -2 \)) to determine \( y \)-values. 4. Plot these points and sketch the upward-opening parabola. --- ### **General Tips for Sketching Parabolas:** 1. **Identify the Vertex:** Locate the point \((h, k)\) where the parabola changes direction. 2. **Determine the Direction:** - If \( a > 0 \), the parabola opens **upwards**. - If \( a < 0 \), it opens **downwards**. 3. **Axis of Symmetry:** Draw a vertical line through the vertex at \( x = h \). 4. **Width of the Parabola:** - \(|a| > 1\): **Narrower** than the standard parabola. - \(|a| < 1\): **Wider** than the standard parabola. - \(|a| = 1\): **Standard** width. 5. **Plot Additional Points:** Choose \( x \)-values on either side of the vertex to find corresponding \( y \)-values. This helps in accurately drawing the shape. 6. **Draw the Parabola:** Use the vertex, axis of symmetry, and plotted points to sketch the U-shaped curve. By following these steps for each function, you can accurately sketch all the given parabolas. If you have graph paper or a graphing tool, it can greatly assist in visualizing these functions.