But before we get ahead of ourselves, let's begin with the parent functions.
Parent functions are "a set of basic functions used as building blocks for more complicated functions." (Mathwords)
Examples of these parent functions are:
As you can see these graphs touch the origin and are not stretched, compressed, or flipped in any way.
But now let's move away from the parent functions. We'll start with something simple like shifting the graph of \[f(x)=x^2\] along the y-axis using the equation \[f(x)=x^2-2\].
Seems pretty strait forward, right? You subtract 2 from \[x^2\] and the y-coordinate of the vertex moves down 2. If you were to add 2, it would move up 2. Simple enough.
Now you may be thinking something along the lines of "If we subtracted 2 from \[x\], why did the graph move to the right?" That's the counter-intuitive part I was talking about. When you subtract from \[x\] the graph moves to the right and when you add to \[x\] the graph moves to the left.
If you would prefer to just memorize the above fact then, by all means, go right ahead. If you would like an explanation, I've got one. Since the graph is only moving along the x-axis the value of the y-coordinate (\[f(x)\]) will always equal \[0\].
Using this information we get that \[0=(x-2)^2\]. Now it comes down to basic Algebra.
This same counter-intuitiveness applies to stretching and compressing. It makes sense that multiplying a function by 2 would make its graph wider and multiplying that same function by \frac{1}{2} would make it more narrow. However, the oposite of this is true. Multiplying a function by 2 makes its graph more narrow and multiplying that same function by \frac{1}{2} would makes it wider.
The best way to understand this is to see it in the actual graph.
As you can see the function \[f(x)=2x^2\] (the green graph) is more narrow than its parent function. This is because the function's outputs increase more quickly than the parent function. The same principle applies with the function \[f(x)=\frac{1}{2}x^2\] (the blue graph). It is wider because it's outputs increase slower than the parent function.
Now we move on to reflections. We will be using a cubic function for this part because it will better illustrate the results of the reflection.
To get a reflection in the x-axis all you have to do is take the function (in this case \[f(x)=2x^3-3x^2-4x \]) and multiply it by -1. It should look like this: \[f(x)=-1(2x^3-3x^2-4x)\]
But what if you want something reflected in the y-axis? All you have to do is have the function be \[f(-x)\]. In this case that would be \[f(-x)= 2(-x)^3-3(-x)^2-4(-x)\]
But what if you want to move it along the x-axis? This is where things get a little counter-intuitive. Before we worry about that, we need to figure out how to move it along the x-axis to begin with.
To move the graph vertically we subtracted 2 after squaring \[x\]. To move the graph horizontally we have to subtract 2 from \[x\] then square it. You can see this in the equation
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Now you may be thinking something along the lines of "If we subtracted 2 from \[x\], why did the graph move to the right?" That's the counter-intuitive part I was talking about. When you subtract from \[x\] the graph moves to the right and when you add to \[x\] the graph moves to the left.
If you would prefer to just memorize the above fact then, by all means, go right ahead. If you would like an explanation, I've got one. Since the graph is only moving along the x-axis the value of the y-coordinate (\[f(x)\]) will always equal \[0\].
Using this information we get that \[0=(x-2)^2\]. Now it comes down to basic Algebra.
This same counter-intuitiveness applies to stretching and compressing. It makes sense that multiplying a function by 2 would make its graph wider and multiplying that same function by \frac{1}{2} would make it more narrow. However, the oposite of this is true. Multiplying a function by 2 makes its graph more narrow and multiplying that same function by \frac{1}{2} would makes it wider.
The best way to understand this is to see it in the actual graph.
As you can see the function \[f(x)=2x^2\] (the green graph) is more narrow than its parent function. This is because the function's outputs increase more quickly than the parent function. The same principle applies with the function \[f(x)=\frac{1}{2}x^2\] (the blue graph). It is wider because it's outputs increase slower than the parent function.Now we move on to reflections. We will be using a cubic function for this part because it will better illustrate the results of the reflection.
To get a reflection in the x-axis all you have to do is take the function (in this case \[f(x)=2x^3-3x^2-4x \]) and multiply it by -1. It should look like this: \[f(x)=-1(2x^3-3x^2-4x)\]
But what if you want something reflected in the y-axis? All you have to do is have the function be \[f(-x)\]. In this case that would be \[f(-x)= 2(-x)^3-3(-x)^2-4(-x)\]
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