Graph \( y=3(x+1)^{2}-2 \) using transformations.

To graph the quadratic function \( y = 3(x + 1)^2 - 2 \) using transformations, you can follow a step-by-step approach starting from the basic parent function. Here's how you can do it: ### **1. Start with the Parent Function** The parent function for this quadratic equation is: \[ y = x^2 \] ### **2. Apply Transformations in Steps** Transformations alter the parent function by shifting, stretching, or compressing it to obtain the desired graph. We'll apply the transformations one at a time in the following order: #### **a. Horizontal Shift (Translation)** **Transformation:** \( y = (x + 1)^2 \) - **Explanation:** The \( +1 \) inside the parentheses shifts the graph **1 unit to the left**. - **New Vertex:** The original vertex of \( y = x^2 \) is at \( (0, 0) \). After the shift, the vertex moves to \( (-1, 0) \). #### **b. Vertical Stretch (Scaling)** **Transformation:** \( y = 3(x + 1)^2 \) - **Explanation:** The coefficient **3** multiplies the entire function, causing a **vertical stretch** by a factor of 3. This makes the parabola narrower compared to the parent function. - **Effect on Vertex:** The vertex remains at \( (-1, 0) \), but the parabola becomes steeper. #### **c. Vertical Shift (Translation)** **Transformation:** \( y = 3(x + 1)^2 - 2 \) - **Explanation:** The \( -2 \) subtracts from the entire function, shifting the graph **2 units downward**. - **New Vertex:** The vertex moves from \( (-1, 0) \) to \( (-1, -2) \). ### **3. Summary of Transformations** 1. **Start with** \( y = x^2 \) (vertex at \( (0, 0) \)). 2. **Shift left** by 1 unit: \( y = (x + 1)^2 \) (vertex at \( (-1, 0) \)). 3. **Vertical stretch** by a factor of 3: \( y = 3(x + 1)^2 \) (vertex at \( (-1, 0) \), steeper parabola). 4. **Shift downward** by 2 units: \( y = 3(x + 1)^2 - 2 \) (vertex at \( (-1, -2) \)). ### **4. Plotting the Graph** Here's how you can plot the graph based on the transformed equation: 1. **Plot the Vertex:** Start by plotting the vertex at \( (-1, -2) \). 2. **Determine Additional Points:** - Choose x-values around the vertex to find corresponding y-values. - For example: - **When \( x = -1 \):** \[ y = 3(-1 + 1)^2 - 2 = 3(0)^2 - 2 = -2 \] Point: \( (-1, -2) \) (vertex) - **When \( x = 0 \):** \[ y = 3(0 + 1)^2 - 2 = 3(1)^2 - 2 = 3 - 2 = 1 \] Point: \( (0, 1) \) - **When \( x = -2 \):** \[ y = 3(-2 + 1)^2 - 2 = 3(-1)^2 - 2 = 3 - 2 = 1 \] Point: \( (-2, 1) \) - **When \( x = 1 \):** \[ y = 3(1 + 1)^2 - 2 = 3(2)^2 - 2 = 12 - 2 = 10 \] Point: \( (1, 10) \) - **When \( x = -3 \):** \[ y = 3(-3 + 1)^2 - 2 = 3(-2)^2 - 2 = 12 - 2 = 10 \] Point: \( (-3, 10) \) 3. **Draw the Parabola:** - Plot all the calculated points. - Draw a smooth, U-shaped curve passing through these points, ensuring it is narrower than the parent \( y = x^2 \) due to the vertical stretch. - The axis of symmetry is the vertical line passing through the vertex: \( x = -1 \). ### **5. Final Graph Characteristics** - **Vertex:** \( (-1, -2) \) - **Axis of Symmetry:** \( x = -1 \) - **Direction:** Opens **upward** (since the coefficient of \( (x + 1)^2 \) is positive). - **Width:** **Narrower** than the parent function \( y = x^2 \) due to the vertical stretch by factor 3. ### **6. Visual Representation** While I can't provide a visual graph here, following the steps above on graph paper or using graphing software will help you accurately plot \( y = 3(x + 1)^2 - 2 \). Here's a quick summary table of points to plot: | \( x \) | \( y = 3(x + 1)^2 - 2 \) | |---------|--------------------------| | -3 | 10 | | -2 | 1 | | -1 | -2 | | 0 | 1 | | 1 | 10 | Plot these points and sketch the parabola accordingly. ### **7. Example Using Graphing Software** If you prefer using graphing tools, entering the equation \( y = 3(x + 1)^2 - 2 \) into graphing software like Desmos, GeoGebra, or a graphing calculator will display the transformed parabola based on the steps outlined above. --- By understanding and applying these transformations, you can graph a variety of quadratic functions systematically and predict how changes to the equation affect the graph's appearance.