Now that you have mastered parameter changes for parabolas in the form y = x², let's change our focus on sideways parabolas in the form x = y².

The parameter changes for a sideways parabola will be similar to the ones you have already practiced. We will use the form:

x = a (y – k)² + h

Remember – whenever the equation is in this form, and the y-term is being squared, the graph is a parabola that opens left or right.

The Effects of a

• Change the value of the slider for the a parameter.
• What happens when it changes from positive to negative?
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The graph reflects over the y-axis.
• What happens when a becomes more and more positive?
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The parabola appears to get skinnier.
• What happens when a becomes more and more negative?
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The graph is reflected over the y-axis and appears to get skinnier.

The Effects of h and k

• Click the RESET button (the double circular arrows in the top right corner of the interactive).


• Change the value of the slider for the h parameter.


• What happens when h changes from 0 to a positive value?
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The graph is translated (shifted) to the RIGHT.
• What happens when h changes from 0 to a negative value?
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The graph is translated (shifted) to the LEFT.
• Click the RESET button (the double circular arrows in the top right corner of the interactive).
• Change the value of the slider for the k parameter.
• What happens when k changes from 0 to a positive value?
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The graph is translated (shifted) UP.
• What happens when k changes from 0 to a negative value?
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The graph is translated (shifted) DOWN.

Now that you have studied those parameter changes, it's important to note a few details.

The equation is generalized as x = a (y – k)² + h. The h-value is still associated with x and with horizontal translations. When hwas positive, the graph shifted left. When h was negative, the graph shifted right.

Likewise, the k-value is still associated with y and vertical translations. When k was positive, the graph shifted up. When k was negative, the graph shifted down. And the a-value still affects the direction the graph opens (positive to the right and negative to the left) and the width of the parabola. As a gets larger and larger, the parabola appears to get skinnier and skinnier.

Check Your Understanding

1. If you have the equation x + 5 = (y – 9)², where will the vertex of the parabola be?
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The vertex will be at the point (–5, 9).
2. What is the vertex for the parabola defined as x – 200 =
   This text is hidden. Fraction: numerator two, denominator five.
   2 5 (y + 160)²?
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The vertex will be at the point (200, –160).
3. Where will the vertex be for the equation x – 3 = –
   This text is hidden. Fraction: numerator one, denominator five.
   1 5 y²? Which way will the graph open? Why? Will it be narrower or wider than the graph of x = y²? Why?
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This equation is equivalent to x – 3 = –
This text is hidden. Fraction: numerator one, denominator five.
1 5 (y – 0)², so the vertex will be at the point(3, 0). And since the a-value is –
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1 5 , it will open LEFT and appear wider than the graph of x = y².
4. What is the equation of the parabola that opens left with a =
   This text is hidden. Fraction: numerator four, denominator three.
   4 3 and its vertex at the point (15, –12)?
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The equation will be x – 15 =
This text is hidden. Fraction: numerator four, denominator three.
–4 3 (y – (–12))², which would simplify to x – 15 =
This text is hidden. Fraction: numerator four, denominator three.
–4 3 (y + 12)². Notice the a-value is negative since we are told the parabola opens left.