Determine the points of intersection of the given functions.

Question

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: \[ \left( \frac{5 + 3\sqrt{5}}{2}, 10 - 3\sqrt{5} \right) \quad \text{and} \quad \left( \frac{5 - 3\sqrt{5}}{2}, 10 + 3\sqrt{5} \right) \]

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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.