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### Course: Algebra (all content)>Unit 8

Lesson 1: Graphs of absolute value functions

# Scaling & reflecting absolute value functions: equation

The graph of y=k|x| is the graph of y=|x| scaled by a factor of |k|. If k<0, it's also reflected (or "flipped") across the x-axis. In this worked example, we find the equation of an absolute value function from a description of the transformation performed on y=|x|.

## Want to join the conversation?

• Is there any way to tilt or rotate the function? I.e rotate the function by say 45 degrees.
• yes, there is a way, but it requires knowledge of things called parametric equations, trigonometry, and also linear algebra. you'll encounter these later on in your math classes (assuming you take them past grade 10)
• what scaled even mean?
• "Scaled" means the graph rises faster or slower than the standard function of y=|x|. For example, y=2|x| rises twice as fast so the V-shape will be narrower.
Hope this helps.
• The 7 is the slope, right?
• Sort of, but not quite. The function would be f(x) = - 7 | x|. For x ≤ 0, the slope would be 7 with a y intercept of (0,0). For x ≥ 0, the slope would be -7 with a y intercept of (0,0). So you have to consider which part of the graph you are talking about.
• Can u flip it on the y axis as well?
• Yes you can flip it on y axis if the points are on the left or right side of the axis.
• Why do we call it "vertical" scaling? What's the difference between scaling such graphs vertically and horizontally?
• Great question! 😊 When we talk about "vertical" scaling, we're talking about stretching or squishing the graph up or down. This happens when we multiply the whole function (like by a number.)

On the other hand, "horizontal" scaling means stretching or squishing the graph left or right. This happens when we multiply the input by a number.

So, in the video, when we multiply the absolute value function by -7, we're doing a vertical scale (and a reflection because of the negative sign)! - Khanmigo AI, (thought I'd try the handy ai feature!)
• So scaling by a factor of 7 is different from just moving the function up by 7 like in the previous video?
• Yes. Scaling is sort of like stretching the graph vertically, where as moving the function up preserves the shape. If the function is x, scaling it would produce 7x, where as moving it upwards produces x+7. If you graph both functions, you can clearly see the difference.
(1 vote)
• can functions have graphs in a v shape and not be absolute value functions? Like an assymetrical v shape?
• Yes, piecewise functions have a different calculation that determines the shape of each piece.
(1 vote)
• Is there any way to tilt or rotate the function? I.e rotate the function by say 45 degrees.
(1 vote)
• I don't think so. Wouldn't the function then change, because of each input having only one output?
• Confused. Since x is absolute value it is always positive? How can the graph go to the left of zero?
(1 vote)
• The value of "x" doesn't need to be positive. The absolute value turns "x" into a positive. For example, if x=-3, then |-3|=+3. The absolute value is impacting the value of Y, not X.

Hope this helps.