Question

Determine the points of intersection of the given functions.

y = –x2 + 3x + 20

y = –2x + 15

Describe the features of the graphing calculator you would choose and why.

Answer 1

To determine the points of intersection of the two functions given:

1. **Functions Overview**:
- Function 1: \( y = -x^2 + 3x + 20 \) (a quadratic function)
- Function 2: \( y = -2x + 15 \) (a linear function)

2. **Finding Points of Intersection**:
We need to set the two equations equal to each other to find the \( x \)-values at which they intersect:
\[
-x^2 + 3x + 20 = -2x + 15
\]

Rearranging the equation:
\[
-x^2 + 3x + 2x + 20 - 15 = 0
\]
\[
-x^2 + 5x + 5 = 0
\]
Multiplying the whole equation by -1 for simplicity:
\[
x^2 - 5x - 5 = 0
\]

3. **Applying the Quadratic Formula**:
We can solve for \( x \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = -5 \), and \( c = -5 \).
\[
x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-5)}}{2 \cdot 1}
\]
\[
x = \frac{5 \pm \sqrt{25 + 20}}{2}
\]
\[
x = \frac{5 \pm \sqrt{45}}{2}
\]
\[
x = \frac{5 \pm 3\sqrt{5}}{2}
\]

So the two \( x \)-values are:
\[
x_1 = \frac{5 + 3\sqrt{5}}{2}, \quad x_2 = \frac{5 - 3\sqrt{5}}{2}
\]

4. **Finding Corresponding \( y \)-Values**:
To find the \( y \)-values at these \( x \)-values, substitute them back into either original equation. Using the linear equation \( y = -2x + 15 \) for simplicity:

For \( x_1 = \frac{5 + 3\sqrt{5}}{2} \):
\[
y_1 = -2\left(\frac{5 + 3\sqrt{5}}{2}\right) + 15 = - (5 + 3\sqrt{5}) + 15 = 10 - 3\sqrt{5}
\]

For \( x_2 = \frac{5 - 3\sqrt{5}}{2} \):
\[
y_2 = -2\left(\frac{5 - 3\sqrt{5}}{2}\right) + 15 = - (5 - 3\sqrt{5}) + 15 = 10 + 3\sqrt{5}
\]

5. **Final Intersection Points**:
The points of intersection are:
\[
\left( \frac{5 + 3\sqrt{5}}{2}, 10 - 3\sqrt{5} \right) \quad \text{and} \quad \left( \frac{5 - 3\sqrt{5}}{2}, 10 + 3\sqrt{5} \right)
\]

---

### Graphing Calculator Features

If I were to choose a graphing calculator to visualize and solve these functions, some important features to consider would be:

1. **Graphing Capabilities**: The capability to graph multiple functions simultaneously, which allows for visual identification of intersection points.

2. **Zoom Features**: Zooming capabilities to focus on specific areas of the graph for more precise analysis or to expand/reduce viewing areas.

3. **Trace Function**: A graphing calculator that allows tracing along the curve to find specific coordinates accurately.

4. **Intersection Calculation**: A built-in feature to calculate intersection points automatically would be very useful to check manual calculations and ensure accuracy.

5. **Algebraic Solver**: An option for solving quadratic equations or systems of equations would help find roots and points of intersection easily.

6. **User Interface**: A clear, user-friendly interface that enhances the ease of inputting functions and reading results without frustration.

7. **Data/Statistics Analysis**: Features for further analysis, like derivatives or area calculations, could be beneficial for broader mathematical applications.

A calculator with these features, such as the TI-84 Plus or the Casio fx-9860GII, would be suitable for students and professionals alike.