6. \( y=9 x^{2}-24 x \)

Certainly! Let's analyze the quadratic function you've provided: \[ y = 9x^{2} - 24x \] This is a quadratic equation in the standard form: \[ y = ax^{2} + bx + c \] where: - \( a = 9 \) - \( b = -24 \) - \( c = 0 \) ### 1. **Graphing the Function** The graph of a quadratic function is a **parabola**. Since the coefficient of \( x^{2} \) (which is \( a = 9 \)) is positive, the parabola opens ** upwards**. ### 2. **Finding the Vertex** The **vertex** of the parabola is the highest or lowest point on the graph. For a parabola that opens upwards, the vertex represents the minimum point. The vertex \((h, k)\) can be found using the formula: \[ h = -\frac{b}{2a} \] \[ k = y(h) \] Let's calculate: - \( h = -\frac{-24}{2 \times 9} = \frac{24}{18} = \frac{4}{3} \) - To find \( k \), substitute \( h \) back into the equation: \[ k = 9\left(\frac{4}{3}\right)^{2} - 24\left(\frac{4}{3}\right) = 9 \times \frac{16}{9} - \frac{96}{3} = 16 - 32 = -16 \] **Vertex:** \( \left( \frac{4}{3}, -16 \right) \) ### 3. **Axis of Symmetry** The **axis of symmetry** is a vertical line that passes through the vertex. It has the equation: \[ x = h \] So, \[ x = \frac{4}{3} \] ### 4. **Finding the X-Intercepts (Roots)** To find the **x-intercepts**, set \( y = 0 \) and solve for \( x \): \[ 0 = 9x^{2} - 24x \] Factor out \( 3x \): \[ 0 = 3x(3x - 8) \] Set each factor equal to zero: \[ 3x = 0 \quad \Rightarrow \quad x = 0 \] \[ 3x - 8 = 0 \quad \Rightarrow \quad x = \frac{8}{3} \] **X-Intercepts:** \( (0, 0) \) and \( \left( \frac{8}{3}, 0 \right) \) ### 5. **Y-Intercept** To find the **y-intercept**, set \( x = 0 \): \[ y = 9(0)^{2} - 24(0) = 0 \] **Y-Intercept:** \( (0, 0) \) ### 6. **Completing the Square** To express the quadratic in **vertex form**, which is: \[ y = a(x - h)^{2} + k \] Starting with: \[ y = 9x^{2} - 24x \] Factor out the coefficient of \( x^{2} \): \[ y = 9\left(x^{2} - \frac{24}{9}x\right) = 9\left(x^{2} - \frac{8}{3}x\right) \] Complete the square inside the parentheses: \[ y = 9\left(x^{2} - \frac{8}{3}x + \left(\frac{4}{3}\right)^{2} - \left(\frac{4}{3}\right)^{2}\right) = 9\left(\left(x - \frac{4}{3}\right)^{2} - \frac{16}{9}\right) \] Distribute the 9: \[ y = 9\left(x - \frac{4}{3}\right)^{2} - 16 \] **Vertex Form:** \[ y = 9\left(x - \frac{4}{3}\right)^{2} - 16 \] Which confirms the vertex at \( \left( \frac{4}{3}, -16 \right) \). ### 7. **Summary** - **Vertex:** \( \left( \frac{4}{3}, -16 \right) \) - **Axis of Symmetry:** \( x = \frac{4}{3} \) - **X-Intercepts:** \( (0, 0) \) and \( \left( \frac{8}{3}, 0 \right) \) - **Y-Intercept:** \( (0, 0) \) - **Direction:** Parabola opens upwards - **Vertex Form:** \( y = 9\left(x - \frac{4}{3}\right)^{2} - 16 \) ### 8. **Graph of the Function** Here's a rough sketch of the graph based on the information above: ``` | 10| | | 0|--------*--------*--------*--------*-------- x | (0,0) (4/3, -16) (8/3,0) -10| | ``` - The parabola passes through the points \( (0,0) \), \( \left( \frac{4}{3}, -16 \right) \), and \( \left( \frac{8}{3}, 0 \right) \). - It opens upwards, with the lowest point at the vertex. If you have a specific question or need further analysis related to this equation, feel free to ask!