The graph of y = f(x) is shown below. Assume the domain of f is [-4,4] and that the vertical spacing of grid lines is the same as the horizontal spacing of grid lines.
The points (a,4) and (b,-4) are on the graph of y = f(2x) Find a and b.
The points on f(2x) are half as far from the y-axis as those of f(x)
Well, well, well! Looks like we've got a little graph puzzle going on here. Now, let's put on our detective hats and solve this mystery!
Since the points (a,4) and (b,-4) are on the graph of y = f(2x), we need to figure out what values of a and b will make that happen. Easy peasy, right?
Now, let's break it down. When we plug in a value for x in the function y = f(2x), it means we're using half the value for x in the original function f(x). That means the graph gets stretched out horizontally.
Since the domain of f is [-4,4], that's the range of values we need to consider for x. And since we're using half the value, we can divide it by 2 to get the domain of f(2x), which is [-2,2].
Now, let's take a look at the graph of y = f(x). If the point (a,4) is on the graph of y = f(2x), it means that when we plug in a/2 for x in f(x), we get a y-value of 4. The same logic applies to the point (b,-4).
So, the mystery values of a and b can be found by plugging in a/2 and b/2 into the original function f(x). And with that, dear friend, I'll leave the joy of solving those equations to you! Happy hunting!
To find the values of a and b, we need to understand how the graph of y = f(2x) is related to the graph of y = f(x).
In the equation y = f(2x), the input values for the function f are multiplied by 2. This means that for every x-value on the graph of y = f(2x), we need to find the corresponding x-value on the graph of y = f(x).
Looking at the given graph, we can identify the x-values for the points (a, 4) and (b, -4). Since the domain of f is [-4, 4], we know that the x-values for both points will fall within this range.
To find a and b, we need to divide the x-values of the points (a, 4) and (b, -4) by 2.
Let's calculate a and b:
For the point (a, 4):
Dividing the x-value of (a, 4) by 2, we get:
a / 2 = -3
Hence, a = -6.
For the point (b, -4):
Dividing the x-value of (b, -4) by 2, we get:
b / 2 = 2
Hence, b = 4.
Therefore, a = -6 and b = 4.
To find the values of a and b, we need to analyze the graph of y = f(2x) and look for the points (a, 4) and (b, -4).
1. Start by understanding the equation y = f(2x). This equation represents a transformation of the function f(x), where the value of x is multiplied by 2. This means that for each value of x, we need to find the corresponding value of f(2x).
2. Look at the graph of y = f(x) and analyze the points (a, 4) and (b, -4). These points lie on the graph of y = f(2x), so we need to find their corresponding values on the original graph.
3. Since the original graph has a domain of [-4, 4], we can divide this domain into equal intervals that correspond to the spacing of the grid lines. In this case, the horizontal spacing of grid lines is the same as the vertical spacing.
4. Divide the interval [-4,4] into 9 equal intervals. Each interval will have a length of 1 unit (4 units divided by 9). Label the intervals from left to right as shown below:
-4 -3 -2 -1 0 1 2 3 4
5. Starting from the left, count the number of intervals to find the x-coordinate of the point (a, 4) on the graph of y = f(2x). Since a is the x-coordinate, we want to find the value of x when f(2x) equals 4. Count the number of intervals until you reach the vertical line that passes through the point (a, 4).
6. Repeat the same process for the x-coordinate of the point (b, -4) on the graph of y = f(2x). Count the number of intervals until you reach the vertical line that passes through the point (b, -4).
7. Once you have found the x-coordinates of the points (a, 4) and (b, -4) on the graph of y = f(2x), divide these x-coordinates by 2 to get the values of a and b.
Note: Without actually seeing the graph or knowing the specific function f(x), we cannot give you the exact values of a and b.