Question

When graphing y = 2x2 + 35x + 75, which viewing window would allow you to see all of the intercepts and the minimum as closely as possible?

the x-axis from –20 to 5, and the y-axis from –80 to 80
the x-axis from –80 to 80, and the y-axis from –20 to 5
the x-axis from –5 to 20, and the y-axis from –80 to 80
the x-axis from –80 to 80, and the y-axis from –5 to 20

Answer 1

To determine the best viewing window for the graph of the function \( y = 2x^2 + 35x + 75 \), we need to find the intercepts and the minimum of the quadratic function.

1. **Finding the x-intercepts:**
We set \( y = 0 \) to find the x-intercepts:
\[
2x^2 + 35x + 75 = 0
\]
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
- Here, \( a = 2 \), \( b = 35 \), and \( c = 75 \).
\[
x = \frac{-35 \pm \sqrt{35^2 - 4 \cdot 2 \cdot 75}}{2 \cdot 2}
\]
\[
= \frac{-35 \pm \sqrt{1225 - 600}}{4}
\]
\[
= \frac{-35 \pm \sqrt{625}}{4}
\]
\[
= \frac{-35 \pm 25}{4}
\]
This gives:
- First x-intercept: \( x = \frac{-10}{4} = -2.5 \)
- Second x-intercept: \( x = \frac{-60}{4} = -15 \)

2. **Finding the y-intercept:**
Set \( x = 0 \):
\[
y = 2(0)^2 + 35(0) + 75 = 75
\]
So the y-intercept is \( (0, 75) \).

3. **Finding the vertex (minimum point):**
The x-coordinate of the vertex for the parabola given by \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \):
\[
x = -\frac{35}{2 \cdot 2} = -\frac{35}{4} = -8.75
\]
Substitute \( x = -8.75 \) back into the function to find the y-coordinate:
\[
y = 2(-8.75)^2 + 35(-8.75) + 75
\]
\[
= 2(76.5625) - 306.25 + 75 = 153.125 - 306.25 + 75 = -78.125
\]

4. **Summary of important points:**
- x-intercepts: \( -2.5 \) and \( -15 \)
- y-intercept: \( 75 \)
- Minimum point (vertex): \( (-8.75, -78.125) \)

5. **Choosing the viewing window:**
We need to see:
- From the lowest x at \( -15 \) to slightly above the highest x intercept at \( -2.5 \),
- The minimum point (approximately at \( -8.75 \)) should be between these two points,
- For the y-axis, it needs to cover at least from \( -80 \) to \( 75 \) (covering the minimum and the y-intercept).

Analyzing the options:
- **Option 1:** x: -20 to 5, y: -80 to 80
- **Option 2:** x: -80 to 80, y: -20 to 5
- **Option 3:** x: -5 to 20, y: -80 to 80
- **Option 4:** x: -80 to 80, y: -5 to 20

**The best option** for seeing all intercepts and the minimum closely is:
**the x-axis from –20 to 5, and the y-axis from –80 to 80** (Option 1).

This option captures both x-intercepts, the y-intercept, and the minimum effectively.