Graph the function f(x)=7(x–4)^2+3.
To graph the function f(x) = 7(x – 4)^2 + 3, we can start by identifying key points and the general shape of the graph.
The vertex form of a quadratic function is given by f(x) = a(x – h)^2 + k, where (h, k) represents the vertex of the parabola. In this case, h = 4 and k = 3.
The vertex represents the lowest or highest point of the parabola, depending on the value of a. Since a = 7, the parabola opens upward.
First, let's plot the vertex (4, 3) on a coordinate plane:
Next, let's find the y-intercept by substituting x = 0 into the equation:
f(0) = 7(0 – 4)^2 + 3
f(0) = 7(16) + 3
f(0) = 112 + 3
f(0) = 115
The y-intercept is 115, so we can plot the point (0, 115) on the graph.
Now, let's find two other points on the graph to make it symmetrical. We can choose to substitute x = 1 and x = 7 into the equation:
f(1) = 7(1 – 4)^2 + 3
f(1) = 7(-3)^2 + 3
f(1) = 7(9) + 3
f(1) = 63 + 3
f(1) = 66
The point (1, 66) is another point on the graph.
f(7) = 7(7 – 4)^2 + 3
f(7) = 7(3)^2 + 3
f(7) = 7(9) + 3
f(7) = 63 + 3
f(7) = 66
The point (7, 66) is symmetric to (1, 66) about the vertex.
Using these points, we can sketch the graph of f(x):
The graph is a U-shaped parabola that opens upward and passes through the vertex (4, 3), the y-intercept (0, 115), and the symmetric points (1, 66) and (7, 66).
To graph the function f(x) = 7(x - 4)^2 + 3, we can follow these steps:
1. Determine the vertex: The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. In this case, the vertex will be (4, 3).
2. Find the y-intercept: The y-intercept is the value of f(x) when x = 0. Substitute x = 0 into the equation and calculate the value of y: f(0) = 7(0 - 4)^2 + 3 = 7(-4)^2 + 3 = 7(16) + 3 = 112 + 3 = 115. So, the y-intercept is (0, 115).
3. Determine the axis of symmetry: The axis of symmetry is the vertical line that passes through the vertex of the parabola. In this case, the axis of symmetry is x = 4.
4. Plot the vertex, y-intercept, and axis of symmetry: Plot the point (4, 3) for the vertex, (0, 115) for the y-intercept, and draw the vertical line x = 4 as the axis of symmetry.
5. Determine additional points: To graph the parabola, you can choose some additional values of x and calculate the corresponding values of f(x) using the given equation. Here are a few points you can calculate:
- When x = 3: f(3) = 7(3 - 4)^2 + 3 = 7(-1)^2 + 3 = 7 + 3 = 10. So, you can plot the point (3, 10).
- When x = 5: f(5) = 7(5 - 4)^2 + 3 = 7(1)^2 + 3 = 7 + 3 = 10. So, you can plot the point (5, 10).
- You can choose more points by substituting different x-values into the equation.
6. Plot the points: Plot the additional points you calculated in the previous step on the graph.
7. Draw the parabola: Connect all the plotted points with a smooth curve. The parabola opens upwards because the coefficient of the x^2 term is positive.
After following these steps, you should have the graph of the function f(x) = 7(x - 4)^2 + 3.
To graph the function f(x) = 7(x–4)^2 + 3, we can start by understanding the general shape of the graph. The given function is in the form of a quadratic function, which means it will be a parabola.
Step 1: Find 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 coordinates of the vertex. In our function, a = 7, h = 4, and k = 3. Therefore, the vertex is located at (4, 3).
Step 2: Determine the direction of the parabola.
Since the coefficient of the quadratic term (x^2) is positive (7), the parabola opens upwards.
Step 3: Locate additional key points.
To sketch the graph, we can find a few more points. Choose some values of x and substitute them into the equation to find the corresponding y-values.
When x = 3:
f(3) = 7(3–4)^2 + 3
= 7(–1)^2 + 3
= 7(1) + 3
= 7 + 3
= 10
Thus, we have the point (3, 10).
When x = 5:
f(5) = 7(5–4)^2 + 3
= 7(1)^2 + 3
= 7(1) + 3
= 7 + 3
= 10
Thus, we have the point (5, 10).
Step 4: Plot the points and draw the parabola.
Using the coordinates of the vertex and the additional points we found, (4, 3), (3, 10), and (5, 10), we can plot them on a coordinate plane. The vertex is the lowest point, and the parabola will open upwards from there. Connect the points smoothly to shape the parabolic curve.
The graph should look like an upward, "U" shaped curve with the vertex at (4, 3).
Note: You can also plot more points if you wish to get a more accurate graph.
Complete the sentence with the correct comparative or superlative adverb.
My grandmother is always frustrated by her hearing aids; she claims she hears even with them in.
Which word goes after even?