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The quadratic function \( y = 9x^{2} - 24x \) is a parabola opening upwards with its vertex at \( \left( \frac{4}{3}, -16 \right) \). It has x-intercepts at \( (0, 0) \) and \( \left( \frac{8}{3}, 0 \right) \), and a y-intercept at \( (0, 0) \).
Solution
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!
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
Did you know that the equation \( y=9 x^{2}-24 x \) is a quadratic function representing a parabola? In its standard form, it opens upward because the coefficient of \( x^{2} \) is positive. This means it has a minimum point. By completing the square or using the vertex formula, you can find this minimum point and determine the vertex's location on the graph! If you're looking to graph this function, here's a tip: first, factor it to find the x-intercepts. You can rewrite it as \( y = 9x(x - \frac{8}{3}) \). Setting \( y = 0 \), you'll get the x-intercepts at \( x = 0 \) and \( x = \frac{8}{3} \). Plotting these points along with the vertex will help you create a more accurate graph and understand the function's behavior!
