Draw the graph of the function \( Y=x^{2}-2 x-3 \) From -2 to +5

Certainly! Let's analyze and graph the quadratic function: \[ Y = x^{2} - 2x - 3 \] over the interval \( x = -2 \) to \( x = 5 \). ### 1. Understanding the Function This is a quadratic function of the form \( Y = ax^{2} + bx + c \), where: - \( a = 1 \) (positive, so the parabola opens upwards) - \( b = -2 \) - \( c = -3 \) ### 2. Key Features of the Graph #### a. **Vertex** The vertex of a parabola given by \( Y = ax^{2} + bx + c \) can be found using: \[ x = -\frac{b}{2a} = -\frac{-2}{2 \times 1} = 1 \] Substitute \( x = 1 \) into the equation to find \( Y \): \[ Y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4 \] **Vertex:** \( (1, -4) \) #### b. **Axis of Symmetry** The vertical line that passes through the vertex: \[ x = 1 \] #### c. **Y-intercept** Set \( x = 0 \): \[ Y = 0^2 - 2(0) - 3 = -3 \] **Y-intercept:** \( (0, -3) \) #### d. **X-intercepts (Roots)** Solve \( Y = 0 \): \[ x^2 - 2x - 3 = 0 \] Factor the quadratic: \[ (x - 3)(x + 1) = 0 \] So, \( x = 3 \) and \( x = -1 \). **X-intercepts:** \( (-1, 0) \) and \( (3, 0) \) #### e. **Additional Points** Let's calculate a few more points within the interval \( x = -2 \) to \( x = 5 \): | \( x \) | \( Y = x^{2} - 2x - 3 \) | |-------|-------------------------| | -2 | \( (-2)^2 - 2(-2) - 3 = 4 + 4 - 3 = 5 \) | | -1 | \( (-1)^2 - 2(-1) - 3 = 1 + 2 - 3 = 0 \) | | 0 | \( 0^2 - 2(0) - 3 = -3 \) | | 1 | \( 1^2 - 2(1) - 3 = -4 \) | | 2 | \( 2^2 - 2(2) - 3 = 4 - 4 - 3 = -3 \) | | 3 | \( 3^2 - 2(3) - 3 = 9 - 6 - 3 = 0 \) | | 4 | \( 4^2 - 2(4) - 3 = 16 - 8 - 3 = 5 \) | | 5 | \( 5^2 - 2(5) - 3 = 25 - 10 - 3 = 12 \) | ### 3. Plotting the Graph While I can't draw the graph directly here, I can provide an ASCII representation to give you an idea of the shape. Keep in mind that ASCII plots are limited in precision. ``` Y 12 | * 10 | 8 | 6 | * * 4 | 2 | 0 | * | * -2| -4| V -6| -8| --------------------------------------- -2 -1 0 1 2 3 4 5 X ``` **Legend:** - `*` represents points on the graph. - `V` denotes the vertex at (1, -4). - The graph is a parabola opening upwards with the vertex at the lowest point. ### 4. Description of the Graph - **Parabola Shape:** The graph is a U-shaped curve (since \( a = 1 > 0 \)). - **Vertex:** The lowest point is at \( (1, -4) \). - **Symmetry:** The parabola is symmetric about the line \( x = 1 \). - **Intercepts:** - It crosses the Y-axis at \( (0, -3) \). - It crosses the X-axis at \( (-1, 0) \) and \( (3, 0) \). - **Behavior:** - As \( x \) moves from left to right, \( Y \) decreases to the vertex and then increases. - For \( x = -2 \), \( Y = 5 \) and for \( x = 5 \), \( Y = 12 \). ### 5. Graphing Tips To plot the graph on graph paper or using software: 1. **Draw the Axes:** Create a horizontal \( x \)-axis and a vertical \( Y \)-axis. 2. **Mark the Vertex:** Plot the point \( (1, -4) \). 3. **Plot Intercepts:** Mark the X-intercepts \( (-1, 0) \) and \( (3, 0) \), and the Y-intercept \( (0, -3) \). 4. **Additional Points:** Plot the additional points from the table above to help shape the parabola. 5. **Draw the Parabola:** Connect the points smoothly to form the U-shaped curve. ### 6. Using Graphing Tools For a more accurate and visually appealing graph, consider using one of the following tools: - **Desmos:** [https://www.desmos.com/calculator](https://www.desmos.com/calculator) - **GeoGebra:** [https://www.geogebra.org/graphing](https://www.geogebra.org/graphing) - **Graphing Calculator Apps:** Available on smartphones and computers. Simply enter the function \( Y = x^{2} - 2x - 3 \) and set the viewing window from \( x = -2 \) to \( x = 5 \) to see the graph. ### 7. Example Graph Using Desmos Here's a snapshot of how the graph would look using the Desmos graphing calculator:  *(Note: Since I can't generate live images, you can visit Desmos and enter the equation to view the graph.)* ### 8. Summary - The quadratic function \( Y = x^{2} - 2x - 3 \) forms an upward-opening parabola. - Key points include the vertex at \( (1, -4) \), X-intercepts at \( (-1, 0) \) and \( (3, 0) \), and Y-intercept at \( (0, -3) \). - Plotting these points and connecting them smoothly will give you the complete graph of the function from \( x = -2 \) to \( x = 5 \). Feel free to use graphing software or graph paper to visualize the function more precisely!