To graph the quadratic equation \( y = -x^2 + 3x + 5 \), let's go through the key steps.

### 1. Identify the standard form
The equation is already in the standard form of a quadratic function:
\[ y = ax^2 + bx + c \]
where \( a = -1 \), \( b = 3 \), and \( c = 5 \).

### 2. Determine the direction of the parabola
Since \( a = -1 \) (which is negative), the parabola opens downward.

### 3. Find the vertex
The vertex \( (h, k) \) of a parabola given by \( y = ax^2 + bx + c \) can be found using the formula:
\[ h = -\frac{b}{2a} \]
Substitute \( a = -1 \) and \( b = 3 \):
\[ h = -\frac{3}{2(-1)} = \frac{3}{2} = 1.5 \]
Now, find \( k \) by substituting \( h = 1.5 \) back into the equation:
\[ k = - (1.5)^2 + 3(1.5) + 5 = -2.25 + 4.5 + 5 = 7.25 \]
So, the vertex is \( (1.5, 7.25) \).

### 4. Find the y-intercept
The y-intercept occurs when \( x = 0 \):
\[ y = - (0)^2 + 3(0) + 5 = 5 \]
So, the y-intercept is \( (0, 5) \).

### 5. Find the x-intercepts (if they exist)
Set \( y = 0 \) to find the x-intercepts:
\[ 0 = -x^2 + 3x + 5 \]
This is a quadratic equation that can be solved using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting \( a = -1 \), \( b = 3 \), and \( c = 5 \):
\[ x = \frac{-3 \pm \sqrt{3^2 - 4(-1)(5)}}{2(-1)} = \frac{-3 \pm \sqrt{9 + 20}}{-2} = \frac{-3 \pm \sqrt{29}}{-2} \]
So the x-intercepts are at:
\[ x = \frac{-3 + \sqrt{29}}{-2} \approx -1.192 \quad \text{and} \quad x = \frac{-3 - \sqrt{29}}{-2} \approx 4.192 \]

### 6. Sketch the graph
- The vertex is at \( (1.5, 7.25) \).
- The y-intercept is at \( (0, 5) \).
- The x-intercepts are approximately at \( (-1.192, 0) \) and \( (4.192, 0) \).
- The parabola opens downward.

### Graph Description
The parabola has its peak at \( (1.5, 7.25) \), crosses the y-axis at \( (0, 5) \), and crosses the x-axis at approximately \( x = -1.192 \) and \( x = 4.192 \).

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**Questions:**
1. How is the vertex of a parabola determined?
2. What effect does the coefficient \( a \) have on the direction of a parabola?
3. How do you find the y-intercept of a quadratic function?
4. What is the quadratic formula used for finding x-intercepts?
5. How would the graph change if \( a \) were positive?

**Tip:** The vertex form of a quadratic function, \( y = a(x-h)^2 + k \), is useful for quickly identifying the vertex and the direction in which the parabola opens.

Learn how to graph the quadratic equation y = -x^2 + 3x + 5 step-by-step with detailed solutions. Understand the vertex, direction, y-intercept, and x-intercepts of the parabola. Suitable for high school students.

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