https://file.aiquickdraw.com/mathbot/file/1726312457663ytvrrl6e.jpg

The problem provides the vertex and one x-intercept of a quadratic function, and it asks for the other x-intercept. Here's how we can solve it:

### Given Information:
- Vertex: \((-3, 6)\)
- One x-intercept: \(\left( -\frac{17}{4}, 0 \right)\)

### Step-by-Step Solution:

1. **Standard form of a quadratic function (vertex form):**
   \[
   f(x) = a(x - h)^2 + k
   \]
   where \((h, k)\) is the vertex. So in this case:
   \[
   f(x) = a(x + 3)^2 + 6
   \]
   
2. **Using the x-intercept to find \(a\):**
   We know the point \(\left( -\frac{17}{4}, 0 \right)\) is an x-intercept, so we substitute it into the equation:
   \[
   0 = a\left( -\frac{17}{4} + 3 \right)^2 + 6
   \]
   Simplifying:
   \[
   -\frac{17}{4} + 3 = -\frac{17}{4} + \frac{12}{4} = -\frac{5}{4}
   \]
   So the equation becomes:
   \[
   0 = a\left( -\frac{5}{4} \right)^2 + 6
   \]
   \[
   0 = a \cdot \frac{25}{16} + 6
   \]
   Solving for \(a\):
   \[
   -6 = a \cdot \frac{25}{16}
   \]
   \[
   a = -6 \cdot \frac{16}{25} = -\frac{96}{25}
   \]

3. **Find the other x-intercept:**
   The equation of the quadratic is now:
   \[
   f(x) = -\frac{96}{25}(x + 3)^2 + 6
   \]
   To find the other x-intercept, set \(f(x) = 0\):
   \[
   0 = -\frac{96}{25}(x + 3)^2 + 6
   \]
   \[
   \frac{96}{25}(x + 3)^2 = 6
   \]
   \[
   (x + 3)^2 = \frac{6 \cdot 25}{96} = \frac{150}{96} = \frac{25}{16}
   \]
   Taking the square root:
   \[
   x + 3 = \pm \frac{5}{4}
   \]
   Solving for \(x\):
   \[
   x = -3 \pm \frac{5}{4}
   \]

   - For the positive root: 
     \[
     x = -3 + \frac{5}{4} = -\frac{12}{4} + \frac{5}{4} = -\frac{7}{4}
     \]
   - For the negative root (which is already given as \(-\frac{17}{4}\)).

Thus, the other x-intercept is \(\boxed{\left( -\frac{7}{4}, 0 \right)}\), which corresponds to option **B**.

Would you like a deeper explanation of any of these steps or have more questions?

### Related Questions:
1. What is the significance of the vertex in a quadratic function?
2. How do you determine if a quadratic function opens upwards or downwards?
3. Can you find the axis of symmetry from the equation of a quadratic function?
4. How does the discriminant help in identifying the number of x-intercepts?
5. How do you convert a quadratic function from standard form to vertex form?

### Tip:
The sign of the coefficient \(a\) in a quadratic function indicates whether the parabola opens upwards (positive \(a\)) or downwards (negative \(a\)).

When the quadratic function f is graphed in the xy-plane, where y = f(x), its vertex is (-3, 6). One of the x-intercepts of this graph is (-17/4, 0). What is the other x-intercept of the graph?

This quadratic function problem involves finding the other x-intercept when the vertex and one x-intercept are given. It uses the vertex form of the quadratic equation and basic algebra to solve for the intercept.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation