Write the absolute value equation that reflects over the x-axis, vertically compresses by a factor of 1/3, and translates 6 to the right.
f(x) = |x|
reflect: -|x|
dilate: -|3x|
shift: -|3(x-6)|
Well, well, well! Let's clown around with this equation, shall we?
To reflect over the x-axis, we can just throw a little negative sign in front of the absolute value function. So, we have -|x|.
Now, for the vertical compression. To compress by a factor of 1/3, we can just squeeze that absolute value graph vertically. So our equation becomes -1/3 |x|.
Finally, the translation. To shift 6 to the right, we just need to replace x with (x - 6). So, our final clownish equation is -1/3 |x - 6|.
Voila! There you have it! The equation that reflects over the x-axis, vertically compresses by a factor of 1/3, and translates 6 to the right. Enjoy the show!
To reflect the absolute value equation over the x-axis, we multiply the equation by -1.
To vertically compress the equation by a factor of 1/3, we multiply the equation by 1/3.
To translate the equation 6 to the right, we subtract 6 from the x-value.
The general form of an absolute value equation is y = |x|.
Let's write the equation step-by-step:
1. Reflect over the x-axis: Multiply by -1
y = -|x|
2. Vertically compress by a factor of 1/3: Multiply by 1/3
y = (-1/3)|x|
3. Translate 6 to the right: Subtract 6 from the x-value
y = (-1/3)|x - 6|
To write an absolute value equation that reflects over the x-axis, vertically compresses, and translates 6 to the right, we need to consider the general form of an absolute value equation:
y = a * |(x - h)| + k
Where:
- 'a' represents the vertical compression or stretch factor
- (h, k) represents the vertex of the absolute value graph, which corresponds to the translations along the x and y axes.
Step 1: Reflection over the x-axis
To reflect a graph over the x-axis, negate the y-value, so the equation becomes:
y = -a * |(x - h)| + k
Step 2: Vertical compression by a factor of 1/3
To vertically compress the graph by a factor of 1/3, multiply the value of 'a' by 1/3:
y = (-1/3) * (-a * |(x - h)|) + k
Simplifying, we get:
y = (1/3) * a * |(x - h)| + k
Step 3: Translation to the right by 6 units
To translate the graph 6 units to the right, we subtract 6 from the x-value inside the absolute value function:
y = (1/3) * a * |(x - (h - 6))| + k
Therefore, the required absolute value equation that reflects over the x-axis, vertically compresses by a factor of 1/3, and translates 6 to the right is:
y = (1/3) * a * |(x - (h - 6))| + k