https://file.aiquickdraw.com/mathbot/file/1727800590785kc4ysn0s.jpg

The image shows a quadratic function problem with the equation \( f(x) = -3x^2 + 7x + 5 \). The task asks for the following:

1. **Find the vertex** of the quadratic function (already partially calculated as \(\left( \frac{7}{6}, \frac{109}{12} \right)\)).
2. **Determine whether the graph opens up or down**.

### Answer Explanation:

1. **Vertex**:
   The vertex form of a quadratic function \( f(x) = ax^2 + bx + c \) is given by:
   \[
   x = -\frac{b}{2a}
   \]
   For \( f(x) = -3x^2 + 7x + 5 \):
   - \( a = -3 \)
   - \( b = 7 \)

   Substituting into the vertex formula:
   \[
   x = -\frac{7}{2(-3)} = \frac{7}{6}
   \]
   The \( y \)-coordinate of the vertex can be found by substituting \( x = \frac{7}{6} \) back into the original equation:
   \[
   f\left(\frac{7}{6}\right) = -3\left(\frac{7}{6}\right)^2 + 7\left(\frac{7}{6}\right) + 5 = \frac{109}{12}
   \]
   So, the vertex is \( \left( \frac{7}{6}, \frac{109}{12} \right) \).

2. **Graph opens up or down**:
   The graph of a quadratic function opens up if \( a > 0 \) and opens down if \( a < 0 \).
   Here, \( a = -3 \), which is less than 0, so the graph opens **down**.

Would you like a more detailed explanation on any step or further assistance with the problem?

---

### Related Questions:

1. How do you convert a quadratic equation into vertex form?
2. How do you find the axis of symmetry for a quadratic function?
3. What is the significance of the sign of \( a \) in determining the direction of a parabola?
4. How do you calculate the y-intercept of a quadratic equation?
5. Can a quadratic function have more than one vertex?

### Tip:
Always check the sign of the coefficient \( a \) to quickly determine whether the parabola opens up or down.

Use the quadratic function f(x) = -3x^2 + 7x + 5 to answer the following questions. (a) Find the vertex. (b) Does the graph open up or down?

Learn how to find the vertex of the quadratic function f(x) = -3x^2 + 7x + 5 and determine whether the parabola opens up or down. This solution uses the vertex formula and covers essential algebraic concepts, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation