To determine what Zander can conclude about the function represented by the graph (let's refer to it as \( g(x) \)) and the function \( f(x) = (x + 4)^2 \), we need to analyze the properties of the function \( f(x) \).
-
Vertex: The function \( f(x) = (x + 4)^2 \) is a parabola that opens upwards. Its vertex can be found at the point \((-4, 0)\), as the vertex form of a quadratic function is given by \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex.
-
X-Intercept: The x-intercept of \( f(x) = (x + 4)^2 \) occurs when \( f(x) = 0 \). Setting \( (x + 4)^2 = 0 \) gives \( x = -4 \). Thus, the function has one x-intercept at \( x = -4 \).
-
Y-Intercept: The y-intercept is found by evaluating \( f(0) \):
\[ f(0) = (0 + 4)^2 = 16 \] So the y-intercept is at \( (0, 16) \). -
Range: Since this is an upward-opening parabola, the range of \( f(x) \) is \( [0, \infty) \) because the vertex is at \( (−4, 0) \), and the output values of \( f(x) \) are all equal to or greater than 0.
Now, Zander can come to the following conclusions based on the information we've derived and the properties of the graph of \( g(x) \):
- If the vertex of \( g(x) \) is at \((-4, 0)\), then they have the same vertex.
- If \( g(x) \) also has an x-intercept at \( x = -4\), they share the same x-intercept.
- If the y-intercept of \( g(x) \) is at \( (0, 16)\), they share the same y-intercept.
- If \( g(x) \) also has a range of \([0, \infty)\), they share the same range.
Without information about the properties of the graph, Zander can only conclude what is explicitly true from the function \( f(x)\):
Given the structure of \( f(x) \), the most definitive conclusion Zander can draw is that it has one x-intercept that is the same (both at \( x = -4 \)), assuming the graph also intersects the x-axis at the same point.
Thus, Zander can conclude: They have one x-intercept that is the same.