Question

Using the included equations, compare x-intercept(s), y-intercept(s), and intersecting point(s) of the linear and quadratic functions.

f(x)=(x+3)^2 and g(x) = 2x+6

Responses:

Both Linear and Quadratic share the x-intercept at (-3, 0). They have different y-intercepts; Linear y-intercept is 6 and Quadratic y-intercept is 9. They also share the point (-1, 4).

Both Linear and Quadratic share their x- and y-intercepts. Their x-intercept is -1 and their y-intercept is 4. They intersect at the point (-3, 0)

These graphs have no shared points.

Answer 1

To analyze the functions and find their x-intercepts, y-intercepts, and any points of intersection, let's evaluate both functions given:

1. **Quadratic Function**:
\( f(x) = (x+3)^2 \)

- **Finding the x-intercept(s)**:
Set \( f(x) = 0 \):
\[
(x + 3)^2 = 0
\]
Taking the square root, we get:
\[
x + 3 = 0 \Rightarrow x = -3
\]
So, the x-intercept of \( f(x) \) is at the point: **(-3, 0)**.

- **Finding the y-intercept**:
Set \( x = 0 \):
\[
f(0) = (0 + 3)^2 = 9
\]
Thus, the y-intercept of \( f(x) \) is at the point: **(0, 9)**.

2. **Linear Function**:
\( g(x) = 2x + 6 \)

- **Finding the x-intercept(s)**:
Set \( g(x) = 0 \):
\[
2x + 6 = 0
\]
Solving for \( x \):
\[
2x = -6 \Rightarrow x = -3
\]
So, the x-intercept of \( g(x) \) is also at the point: **(-3, 0)**.

- **Finding the y-intercept**:
Set \( x = 0 \):
\[
g(0) = 2(0) + 6 = 6
\]
Thus, the y-intercept of \( g(x) \) is at the point: **(0, 6)**.

3. **Finding the intersection point(s)**:
We set \( f(x) = g(x) \):
\[
(x + 3)^2 = 2x + 6
\]
Expanding and rearranging:
\[
x^2 + 6x + 9 = 2x + 6
\]
\[
x^2 + 4x + 3 = 0
\]
Factoring gives:
\[
(x + 1)(x + 3) = 0
\]
This provides roots \( x = -1 \) and \( x = -3 \).

- For **x = -1**:
\[
f(-1) = (-1 + 3)^2 = 4 \quad \text{and} \quad g(-1) = 2(-1) + 6 = 4
\]
So, they intersect at: **(-1, 4)**.

4. **Summary of intercepts and intersection**:
- **x-intercept**: Both \( f(x) \) and \( g(x) \) have the same x-intercept at **(-3, 0)**.
- **y-intercept**: \( f(x) \) has a y-intercept at **(0, 9)** while \( g(x) \) has a y-intercept at **(0, 6)**.
- **Intersection Point**: They intersect at **(-1, 4)**.

Therefore, the correct response based on this analysis is:

**Both Linear and Quadratic share the x-intercept at (-3, 0). They have different y-intercepts; Linear y-intercept is 6 and Quadratic y-intercept is 9. They also share the point (-1, 4).**