The Family of Parabolas

The function \(y=x^2\)

The family of parabolas \(y=xยฒ+c\)

The family of parabolas \(y=(x-p)ยฒ\)

The family of parabolas \(y=(x-p)ยฒ+k\)

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Test yourself on forms of parabolas!

einstein

What is the value of y for the function?

\( y=x^2 \)

of the point \( x=2 \)?

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The Family of Parabolas

Meet the most basic quadratic function:
y=X2y=X^2
In this function:
b=0ย ,c=0ย ,a=1b=0~, c=0~, a=1

*Relevant illustration in Word file*

The function is a smiling minimum function and its vertex is - (0,0)(0,0)
The axis of symmetry of this function is X=0X=0.
The increasing interval of the function โ€“ all the XX values where the function is increasing are: X>0X>0
The decreasing interval of the function โ€“ all the XX values where the function is decreasing are: X<0X<0
Positive interval: all XX except 00 โ€“ you can see in the graph that the entire function is above the XX axis
Negative interval: none. The entire function is above the XX axis.
Let's continue to a similar function from the same family:
y=โˆ’x2y=-x^2
In this function:
a=โˆ’1,ย b=0,ย c=0a=-1 ,~b=0 ,~c=0

*Relevant illustration in the Word file*

The function is a sad maximum function and its vertex is (0,0)(0,0)
The axis of symmetry of this function is X=0X=0.
Increasing interval: X<0X<0
Decreasing interval: X>0X>0
Positive interval: None. The entire function is below the XX axis.
Negative interval: All XX except X=0X=0
Let's continue with another function from the same family:
y=ax2y=ax^2
In this function:
a=someย number,ย b=0,ย c=0a=some~number,~b=0,~ c=0 *Relevant illustration in Word file*

Its vertex is- (0,0)(0,0)
The axis of symmetry of this function is X=0X=0.

The larger aa is: the parabola will have a smaller opening โ€“ closer to its axis of symmetry.
The smaller aa is: the parabola will have a larger opening โ€“ farther from its axis of symmetry.

The function from this family has no horizontal or vertical shift because in every function from this family the vertex is (0,0)(0,0)


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The family of parabolas y=x2+cy=x^2+c
The basic quadratic function โ€“ with the addition of cc
In this function:
cc โ€“ represents the intersection point of the function with the YY axis.
The meaning of cc is a vertical shift up or down of the function from the vertex (0,0)(0,0).
If cc is positive โ€“ the function will shift vertically up by the number of steps indicated by cc.
If cc is negative โ€“ the function will shift vertically down by the number of steps indicated by cc.

This function describes only vertical shifts up and down according to cc
Let's see an example:
y=4x2+7y=4x^2+7

In this function -
C=7C=7
this means that the intersection point of the function with the YY axis is 77.
And actually โ€“ the function moves vertically 77 steps up.
*Illustration in Word file*

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The family of parabolas \(y=(x-p)^2\)

In this family, we are given a quadratic function that clearly shows us how the function moves horizontally โ€“ how many steps it needs to move right or left.
PP represents the number of steps the function will move horizontally โ€“ right or left.
If PP is positive โ€“ (there is a minus in the equation) โ€“ the function will move PP steps to the right.
If PP is negative โ€“ (and as a result, there is a plus in the equation because minus times minus equals plus) โ€“ the function will move PP steps to the left.
Let's see an example:
ย ย ย Y=(Xโˆ’6)2ย ย  ย Y=(X-6)^2

This function moves from the vertex (0,0)(0,0),66 steps to the right
Let's see this in the figure: *relevant figure in the Word file*

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The family of parabolas \(y=(x-p)^2+k \)

In this quadratic function, we can see a combination of horizontal and vertical shifts:
KK: Determines the number of steps and the direction the function will move vertically โ€“ up or down.
KK positive โ€“ shift up, KK negative โ€“ shift down.
PP: Determines the number of steps and the direction the function will move horizontally โ€“ right or left.

Let's look at an example of combining two translations together:
For example, in the function:
y=(x+2)2+5y=(x+2)^2+5

The changes will be:
According to P=โˆ’2P=-2 : the parabola will move 22 steps to the left.
According to K=5K=5 : the parabola will move 55 steps up.

Let's see this in the illustration: *Relevant illustration in the Word file*


We can see that the vertex of the parabola is:
(โˆ’2,5)(-2,5)

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