To analyze the functions and find their x-intercepts, y-intercepts, and any points of intersection, let's evaluate both functions given:
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Quadratic Function: \( f(x) = (x+3)^2 \)
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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).
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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).
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Linear Function: \( g(x) = 2x + 6 \)
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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).
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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).
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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).
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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).