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

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

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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. To sketch the given parabolas, follow these steps for each function: 1. **Identify the Vertex** $$(h, k)$$ from the vertex form $$ y = a(x - h)^2 + k $$. 2. **Determine the Direction** based on the sign of $$ a $$: - If $$ a > 0 $$, the parabola opens upwards. - If $$ a < 0 $$, it opens downwards. 3. **Draw the Axis of Symmetry** at $$ x = h $$. 4. **Plot Additional Points** by choosing $$ x $$-values on either side of the vertex and calculating corresponding $$ y $$-values. 5. **Sketch the Parabola** using the vertex, axis of symmetry, and plotted points. Apply these steps to each of the six functions to accurately sketch their graphs.

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