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You are given two quadratic functions to analyze:

1. \( y = -x^2 + 9x - 14 \)
2. \( y = x^2 + 14x + 40 \)

I will walk through the first function, \( y = -x^2 + 9x - 14 \), to fill out the chart. Let me break down each required element:

### 1. **Orientation (Concavity)**:
   - The coefficient of \( x^2 \) is negative (\( -1 \)), so the parabola opens **downward** (concave down).

### 2. **Minimum or Maximum**:
   - Since the parabola opens downward, it has a **maximum** point.

### 3. **X-Intercepts** (Solutions of \( y = 0 \)):
   - To find the x-intercepts, solve \( -x^2 + 9x - 14 = 0 \).
   - Use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = -1 \), \( b = 9 \), and \( c = -14 \).
     \[
     x = \frac{-9 \pm \sqrt{9^2 - 4(-1)(-14)}}{2(-1)}
     \]
     \[
     x = \frac{-9 \pm \sqrt{81 - 56}}{-2}
     \]
     \[
     x = \frac{-9 \pm \sqrt{25}}{-2}
     \]
     \[
     x = \frac{-9 \pm 5}{-2}
     \]
     So, the two solutions are:
     \[
     x_1 = \frac{-9 + 5}{-2} = 2 \quad \text{and} \quad x_2 = \frac{-9 - 5}{-2} = 7
     \]
     Therefore, the x-intercepts are \( x = 2 \) and \( x = 7 \).

### 4. **Y-Intercept** (Substitute \( x = 0 \) into the equation):
   - Substituting \( x = 0 \) into \( y = -x^2 + 9x - 14 \):
     \[
     y = -(0)^2 + 9(0) - 14 = -14
     \]
     So, the y-intercept is \( y = -14 \).

### 5. **Vertex**:
   - The x-coordinate of the vertex is given by \( x = -\frac{b}{2a} \).
     \[
     x = -\frac{9}{2(-1)} = \frac{9}{2}
     \]
   - Substituting \( x = \frac{9}{2} \) into the equation to find the y-coordinate:
     \[
     y = -\left( \frac{9}{2} \right)^2 + 9\left( \frac{9}{2} \right) - 14
     \]
     \[
     y = -\frac{81}{4} + \frac{81}{2} - 14
     \]
     Converting to a common denominator:
     \[
     y = -\frac{81}{4} + \frac{162}{4} - \frac{56}{4}
     \]
     \[
     y = \frac{-81 + 162 - 56}{4} = \frac{25}{4}
     \]
     So, the vertex is \( \left( \frac{9}{2}, \frac{25}{4} \right) \).

This completes the analysis of the first function.

Would you like me to work on the second equation or provide more details about any of the steps? 

### 5 Relative Questions:
1. How do you calculate the vertex for a quadratic function?
2. Why does the sign of the coefficient of \( x^2 \) determine the concavity?
3. How do you find the x-intercepts using the quadratic formula?
4. What is the significance of the y-intercept in a quadratic graph?
5. How does the vertex relate to the minimum or maximum of a parabola?

### Tip:
Always remember that the quadratic formula works for any quadratic equation, even when the discriminant is zero or negative.

Select one of the following quadratic functions to demonstrate your knowledge: y = -x^2 + 9x - 14 or y = x^2 + 14x + 40. Identify orientation (concavity), minimum or maximum point, x-intercepts, y-intercept, and vertex.

Explore how to analyze the quadratic function y = -x^2 + 9x - 14 by identifying its concavity, intercepts, and vertex. This step-by-step guide walks through solving for key characteristics using the quadratic and vertex formulas. Ideal for students in Grades 9-11.

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