f(x) = (x - 5)2 - 3
x = 13/2
(5,-3)
To determine the graph of the function f(x) = (x - 5)^2 - 3, we can follow a few steps:
Step 1: Identify the vertex of the quadratic function.
The vertex form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) represents the vertex. In our given function, it is clear that h = 5 and k = -3, so the vertex is (5, -3).
Step 2: Determine the axis of symmetry.
The axis of symmetry is a vertical line that passes through the vertex of a parabola. For a quadratic function in the vertex form f(x) = a(x - h)^2 + k, the axis of symmetry is x = h. In our case, the axis of symmetry is x = 5.
Step 3: Find the y-intercept.
To locate the y-intercept, we set x to 0 and calculate the corresponding y-value. Plugging x = 0 into the function f(x), we get f(0) = (0 - 5)^2 - 3 = 22. Therefore, the y-intercept is (0, 22).
Step 4: Plot additional points.
We can choose a few more x-values and plug them into the function to find their corresponding y-values. For example:
- If x = 3, f(3) = (3 - 5)^2 - 3 = 1.
- If x = 7, f(7) = (7 - 5)^2 - 3 = 1.
Plotting these points on a graph, we have:
- Vertex: (5, -3)
- Y-intercept: (0, 22)
- Additional points: (3, 1) and (7, 1)
Step 5: Sketch the graph.
Using the information we have gathered, we can sketch the graph of the quadratic function f(x) = (x - 5)^2 - 3. The graph will be a downward-opening parabola with the vertex at (5, -3), passing through the point (0, 22), and having additional points at (3, 1) and (7, 1).