Answer:
The graph should show a downward-opening parabola with the vertex at (4, 5) and the axis of symmetry as a vertical line passing through x = 4. The parabola will touch the x-axis at x = 2 and x = 6, and its y-coordinate will reach the maximum value of 5 at the vertex (4, 5).
Explanation:
To graph the function g(x) = -5/4(x - 4)^2 + 5 and find the vertex, axis of symmetry, and the maximum or minimum value, follow these steps:
Step 1: Identify the vertex form of the function.
The given function is already in vertex form, which is y = a(x - h)^2 + k, where (h, k) is the vertex.
In this case, a = -5/4, h = 4, and k = 5.
Step 2: Find the vertex and axis of symmetry.
The vertex of the parabola is (h, k), and the axis of symmetry is the vertical line x = h.
Vertex: (h, k) = (4, 5)
Axis of symmetry: x = 4
Step 3: Determine the type of parabola (upward or downward).
The coefficient of the (x - h)^2 term is a = -5/4. Since this value is negative, the parabola will open downwards.
Step 4: Find the maximum value (in the case of a downward-opening parabola).
Since the parabola opens downwards, the vertex is the maximum point on the graph, and its y-coordinate (k) represents the maximum value.
Maximum value: 5
Step 5: Plot the vertex and draw the axis of symmetry.
On the graph, plot the point (4, 5) as the vertex. Then, draw a dashed vertical line at x = 4 to represent the axis of symmetry.
Step 6: Plot additional points and draw the graph.
To draw the graph, choose some x-values on both sides of the axis of symmetry and calculate the corresponding y-values using the function g(x) = -5/4(x - 4)^2 + 5.
For example:
- When x = 2, g(2) = -5/4(2 - 4)^2 + 5 = -5/4(-2)^2 + 5 = -5/4(4) + 5 = -5 + 5 = 0
- When x = 3, g(3) = -5/4(3 - 4)^2 + 5 = -5/4(-1)^2 + 5 = -5/4 + 5 = 5 - 5/4 = 4.75 (approx)
- When x = 5, g(5) = -5/4(5 - 4)^2 + 5 = -5/4(1)^2 + 5 = -5/4 + 5 = 5 - 5/4 = 4.75 (approx)
- When x = 6, g(6) = -5/4(6 - 4)^2 + 5 = -5/4(2)^2 + 5 = -5/4(4) + 5 = -5 + 5 = 0
Now, plot these points (2, 0), (3, 4.75), (5, 4.75), and (6, 0) on the graph.
Step 7: Draw the parabola.
Finally, draw a smooth curve through the plotted points to represent the graph of the function.