For each function, identify the horizontal translation of the parent function, f(x)=x^2. Then graph the function.
19. y=(x-4)^2
I need to figure out how to solve this problem, so steps would be appreciative! PLEASE don't leave websites for me to check out, I guarantee you I've seen them, I just don't understand. Thanks so much :)
Well, well, well, let's have some fun with this math problem! You have the parent function f(x) = x^2, and you want to find the horizontal translation of the function y = (x - 4)^2. Here's how we can do it:
Step 1: Start with the parent function f(x) = x^2.
Step 2: Look at the given function y = (x - 4)^2. Notice the (x - 4) part inside the parentheses.
Step 3: The number inside the parentheses tells us the horizontal translation. In this case, it's 4. "Why?", you ask. Well, when x gets subtracted by 4, it shifts the graph 4 units to the right. So, the function y = (x - 4)^2 is what you get when you move the graph of f(x) = x^2 to the right by 4 units.
Step 4: Time to graph the function. Start with the parent function f(x) = x^2. Then, take each point on the graph and shift it 4 units to the right. Easy-peasy, right?
Remember, math can be a bit of a clown sometimes, but with a little humor, it becomes less scary. Now, go ahead and give it a try!
To identify the horizontal translation of the parent function f(x) = x^2 in the given function y = (x - 4)^2, we need to find the value of the constant term in the transformation equation.
Step 1: Compare the given function with the parent function f(x) = x^2.
The function y = (x - 4)^2 is an example of vertex form of the quadratic equation.
The vertex form of the quadratic equation is given by y = a(x - h)^2 + k, where (h, k) represents the vertex.
Step 2: Compare the constant term in the given function to the vertex form of the quadratic equation.
In the given function y = (x - 4)^2, the constant term is (4)^2.
Step 3: Observe that the constant term, (4)^2, represents the horizontal translation of the parent function.
So, the horizontal translation of the parent function f(x) = x^2 in the function y = (x - 4)^2 is 4 units to the right.
To graph the function, follow these steps:
Step 1: Plot the vertex of the parabola.
The vertex of the parabola is (h, k), which is (4, 0) in this case. Plot the point (4, 0) on the coordinate plane.
Step 2: Determine two other points on the graph of the parabola.
To find two additional points, substitute different values of x into the equation y = (x - 4)^2 and calculate the corresponding y-values.
For example, when x = 3:
y = (3 - 4)^2
y = (-1)^2
y = 1
So, another point on the graph is (3, 1).
Similarly, when x = 5:
y = (5 - 4)^2
y = (1)^2
y = 1
Another point on the graph is (5, 1).
Step 3: Plot the two additional points on the graph.
Plot the points (3, 1) and (5, 1) on the coordinate plane.
Step 4: Draw the parabolic curve passing through the vertex and the two additional points.
Connect the three plotted points (vertex and two additional points) smoothly to form a parabolic curve.
Step 5: Label the graph.
Label the vertex as (4, 0) and any other relevant points if necessary.
This completes the graph of the function y = (x - 4)^2 with a horizontal translation of 4 units to the right.
To identify the horizontal translation of the parent function f(x) = x^2 and graph the function y = (x - 4)^2, you can follow these steps:
Step 1: Start with the parent function f(x) = x^2.
This is the basic parabola, opening upward, with the vertex located at (0, 0).
Step 2: Identify the horizontal translation.
The expression (x - 4) in the function y = (x - 4)^2 represents a horizontal translation. The number 4 inside the parentheses indicates that the graph will be shifted 4 units to the right.
Step 3: Find the new vertex.
To find the new vertex, you need to shift the original vertex of (0, 0) horizontally by 4 units to the right. So, add 4 to the x-coordinate of the original vertex to get the new vertex. In this case, the new vertex is (4, 0).
Step 4: Plot the new vertex.
On your graph, locate the point (4, 0). This will be the new vertex of the function y = (x - 4)^2.
Step 5: Determine the shape of the parabola.
Since the coefficient of x^2 in the function y = (x - 4)^2 is positive, the parabola opens upward, just like the parent function f(x) = x^2.
Step 6: Plot additional points.
Choose some additional x-values, substitute them into the function, and calculate the corresponding y-values. For example, if you choose x = 3, substitute it into the equation: y = (3 - 4)^2 = (-1)^2 = 1. So, you have the point (3, 1). You can choose more x-values and calculate their corresponding y-values to plot additional points.
Step 7: Connect the plotted points.
Using all the plotted points, draw a smooth curve that passes through them. This curve represents the graph of the function y = (x - 4)^2.
By following these steps, you can identify the horizontal translation and graph the given function y = (x - 4)^2 from the parent function f(x) = x^2.
Think of moving the y-axis 4 units to the right. All the x distances would be 4 less than they are now.
So, y = (x-4)^2 is just the parabola y=x^2, moved 4 units to the right!
replacing x by (x-h) moves the graph h units to the right.
you can see this here:
http://www.wolframalpha.com/input/?i=plot+y%3Dx%5E2,+y%3D(x-4)%5E2,+-4%3C%3Dx%3C%3D10