Answer :
To analyze the graphs of the quadratic functions \( f(x) \) and \( g(x) \), let's first examine the function \( f(x) = -(x+2)^2 + 2 \) in detail.
1. Intercepts:
- For the \( x \)-intercepts of \( f(x) \), we set \( f(x) = 0 \):
[tex]\[ 0 = -(x+2)^2 + 2 \][/tex]
Solving for \( x \):
[tex]\[ (x+2)^2 = 2 \][/tex]
[tex]\[ x + 2 = \pm \sqrt{2} \][/tex]
[tex]\[ x = -2 \pm \sqrt{2} \][/tex]
Therefore, \( f(x) \) intersects the \( x \)-axis at \( x = -2 + \sqrt{2} \) and \( x = -2 - \sqrt{2} \).
- For the \( y \)-intercept, we set \( x = 0 \):
[tex]\[ f(0) = -(0+2)^2 + 2 = -4 + 2 = -2 \][/tex]
Thus, the \( y \)-intercept of \( f(x) \) is \(-2\).
2. End Behavior:
- Since \( f(x) \) is a downward-opening parabola (because the coefficient of the square term is negative), as \( x \to \infty \) or \( x \to -\infty \), \( f(x) \to -\infty \).
3. Line of Symmetry:
- The vertex form of the parabola \( f(x) = -(x+2)^2 + 2 \) indicates that the vertex is at \( (-2, 2) \). The line of symmetry is the vertical line passing through the \( x \)-coordinate of the vertex, which is \( x = -2 \).
Now, since we know these details about \( f(x) \), we evaluate the possible statements:
1. "The graphs of \( f(x) \) and \( g(x) \) do not intersect the \( x \)-axis." This statement is incorrect because we have shown that \( f(x) \) does intersect the \( x \)-axis.
2. "The graphs of \( f(x) \) and \( g(x) \) have the same end behavior." Since \( f(x) \) ends as \( x \to \infty \) or \( x \to -\infty \) both go to \( -\infty \), this would be true for any function with a leading negative coefficient.
3. "The \( y \)-intercepts for the graphs of \( f(x) \) and \( g(x) \) are the same." This requires knowing the \( y \)-intercept of \( g(x) \), which isn’t provided.
4. "The lines of symmetry for the graphs of \( f(x) \) and \( g(x) \) are the same." For this to be true, \( g(x) \) must also have a line of symmetry at \( x = -2 \).
Given the information, the most accurate statement is:
- The graphs of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same end behavior.
1. Intercepts:
- For the \( x \)-intercepts of \( f(x) \), we set \( f(x) = 0 \):
[tex]\[ 0 = -(x+2)^2 + 2 \][/tex]
Solving for \( x \):
[tex]\[ (x+2)^2 = 2 \][/tex]
[tex]\[ x + 2 = \pm \sqrt{2} \][/tex]
[tex]\[ x = -2 \pm \sqrt{2} \][/tex]
Therefore, \( f(x) \) intersects the \( x \)-axis at \( x = -2 + \sqrt{2} \) and \( x = -2 - \sqrt{2} \).
- For the \( y \)-intercept, we set \( x = 0 \):
[tex]\[ f(0) = -(0+2)^2 + 2 = -4 + 2 = -2 \][/tex]
Thus, the \( y \)-intercept of \( f(x) \) is \(-2\).
2. End Behavior:
- Since \( f(x) \) is a downward-opening parabola (because the coefficient of the square term is negative), as \( x \to \infty \) or \( x \to -\infty \), \( f(x) \to -\infty \).
3. Line of Symmetry:
- The vertex form of the parabola \( f(x) = -(x+2)^2 + 2 \) indicates that the vertex is at \( (-2, 2) \). The line of symmetry is the vertical line passing through the \( x \)-coordinate of the vertex, which is \( x = -2 \).
Now, since we know these details about \( f(x) \), we evaluate the possible statements:
1. "The graphs of \( f(x) \) and \( g(x) \) do not intersect the \( x \)-axis." This statement is incorrect because we have shown that \( f(x) \) does intersect the \( x \)-axis.
2. "The graphs of \( f(x) \) and \( g(x) \) have the same end behavior." Since \( f(x) \) ends as \( x \to \infty \) or \( x \to -\infty \) both go to \( -\infty \), this would be true for any function with a leading negative coefficient.
3. "The \( y \)-intercepts for the graphs of \( f(x) \) and \( g(x) \) are the same." This requires knowing the \( y \)-intercept of \( g(x) \), which isn’t provided.
4. "The lines of symmetry for the graphs of \( f(x) \) and \( g(x) \) are the same." For this to be true, \( g(x) \) must also have a line of symmetry at \( x = -2 \).
Given the information, the most accurate statement is:
- The graphs of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have the same end behavior.
