Just che king to see if I did this right.
f(x)=(x-4)^2+6
Graph f(x) I graphed 1quadrant 4 up 6
Identify vertex (4,6)
Identify intervals f(x) is increasing and decreasing
D [-00,6) I (6,00]
Checking to see if I figured this right. Thanks
It seems like you have made a few errors in your explanation. Let's go through the steps to graph the function f(x) = (x-4)^2 + 6 correctly and identify the intervals where f(x) is increasing and decreasing.
To graph the function f(x) = (x-4)^2 + 6, follow these steps:
1. Start by identifying the vertex of the parabola. The vertex form of a quadratic function is given by f(x) = a(x-h)^2 + k, where (h, k) represents the vertex.
In this case, (h, k) = (4, 6). So, the vertex of the parabola is (4, 6).
2. Plot the vertex on the coordinate plane. In this case, the vertex is at (4, 6).
3. Determine the direction of the parabola. Since the coefficient of the x^2 term is positive (a = 1), the parabola opens upwards.
4. Find additional points to complete the graph. Choose a few x-values on both sides of the vertex and substitute them into the function to find the corresponding y-values. For example, you can choose x = 3, 5, and 6.
When x = 3:
f(3) = (3-4)^2 + 6
= (-1)^2 + 6
= 1 + 6
= 7
So, when x = 3, y = 7. Plot the point (3, 7) on the graph.
When x = 5:
f(5) = (5-4)^2 + 6
= (1)^2 + 6
= 1 + 6
= 7
So, when x = 5, y = 7. Plot the point (5, 7) on the graph.
When x = 6:
f(6) = (6-4)^2 + 6
= (2)^2 + 6
= 4 + 6
= 10
So, when x = 6, y = 10. Plot the point (6, 10) on the graph.
5. Connect the plotted points to form the graph of the function. You will notice that it is a U-shaped curve opening upwards.
Now, let's identify the intervals where f(x) is increasing and decreasing:
To determine the intervals of increase and decrease, analyze the slope of the function. Since f(x) = (x-4)^2 + 6 is a quadratic function opening upwards, the function is increasing for all x-values greater than the vertex x-coordinate (4) and decreasing for all x-values less than the vertex x-coordinate (4).
Therefore, the interval where f(x) is increasing is (4, ∞) and the interval where f(x) is decreasing is (-∞, 4).
I hope this explanation helps you understand the graphing process and the intervals of increase and decrease for the function f(x) = (x-4)^2 + 6. Feel free to ask if you have any further questions!