Sketch the complete graph of the following function.

Question

y equals 2 x squared plus 4 x minus 9
Question content area bottom left
Part 1
Use the graphing tool to graph the function.

Answer 1

To graph the function \( y = 2x^2 + 4x - 9 \), we should follow these steps:

Step 1: Identify the Components of the Quadratic Equation

The function is in the standard form \( y = ax^2 + bx + c \) where:

• \( a = 2 \)
• \( b = 4 \)
• \( c = -9 \)

Step 2: Find the Vertex

The vertex of a parabola given by a quadratic function can be found using the formula: \[ x = -\frac{b}{2a} \] Substituting in the values: \[ x = -\frac{4}{2 \cdot 2} = -\frac{4}{4} = -1 \]

Next, substitute \( x = -1 \) back into the original equation to find the \( y \)-coordinate of the vertex: \[ y = 2(-1)^2 + 4(-1) - 9 = 2(1) - 4 - 9 = 2 - 4 - 9 = -11 \] Thus, the vertex is \( (-1, -11) \).

Step 3: Find the Y-Intercept

The y-intercept occurs when \( x = 0 \): \[ y = 2(0)^2 + 4(0) - 9 = -9 \] So, the y-intercept is \( (0, -9) \).

Step 4: Find the X-Intercepts

To find the x-intercepts, set \( y = 0 \): \[ 2x^2 + 4x - 9 = 0 \] Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \): \[ x = \frac{-4 \pm \sqrt{4^2 - 4(2)(-9)}}{2(2)} = \frac{-4 \pm \sqrt{16 + 72}}{4} = \frac{-4 \pm \sqrt{88}}{4} = \frac{-4 \pm 2\sqrt{22}}{4} = \frac{-2 \pm \sqrt{22}}{2} \] Thus, the x-intercepts are: \[ x = \frac{-2 + \sqrt{22}}{2}, \quad x = \frac{-2 - \sqrt{22}}{2} \]

Step 5: Sketch the Graph

1. Plot the vertex at \( (-1, -11) \).
2. Plot the y-intercept at \( (0, -9) \).
3. Plot the x-intercepts calculated above.
4. Since the coefficient of \( x^2 \) is positive (2), the parabola opens upwards.
5. Draw the axis of symmetry which is the vertical line \( x = -1 \).

Summary of Points:

• Vertex: \( (-1, -11) \)
• Y-Intercept: \( (0, -9) \)
• X-Intercepts: Approx. \( (-2.9, 0) \) and \( (0.9, 0) \) based on calculations.

Final Graph

You can graph the function using graphing tools or software where you plot the points mentioned and sketch the parabola that opens upward crossing through those points.

This is how you would construct a graph of \( y = 2x^2 + 4x - 9 \). If you have a specific graphing tool, you can input the equation, and it will generate the graph for you.