Quadractic Relationships

Introducing the Parabola

What is a parabola ?

The image that you see to your right is an example of a parabola. A parabola is a curve where any point is at an equal distance from a separate point. Parabolas can either open up or down.

Example

For example, say you wanted to measure how high a soccer ball was kicked after a certain amount of time. You could use the x-axis to measure the time. and the y-axis to measure the height. As you can see in the example below, from the point where the soccer player kicked the ball, according to the time ( x-axis ) you can measure how high it would be in the air.

Terms

Like in all units in math, parabolas have a few terms you have to know. You need to know the vertex, y-intercept, zeros (x-intercepts), optimal value, and the axis of symmetry ( AOS).
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Definitions

Vertex-The maximum or minimum point on the graph. It is the point where the graph changes direction.

Y-intercepts- The y-intercept is the point on the y- axis where the parabola pass through the y-axis.

Zeros- The zeros are in other words, the x- intercepts. This is the point on the graph where the parabola passes through a point on the x-axis.

Optimal value- The highest or lowest value the graph reaches.

Axis of symmetry- The vertical line that separates the parabola in two equal halves.

The effect of a in y=ax(x)

Note: x(x) represents x squared. To begin, the basic step pattern is y=x(x). This means that if it goes up 1 time then it goes over 2 times. if it goes up 2 times, it goes over 4 times and so on and so forth. And this step pattern is used to find the points of the parabola. Now this y=ax(x) is still a step pattern, just modified. Now what this a does is this equation is that it stretches the "U" shape either thicker or thinner depending on what a is. as you can see in the example below, by just multiplying everything by 0.5, it gave me a wider parabola. Futhermore, the higher the number is, the skinner the parabola will be. In conclusion, the a represents the Vertical Stretch.
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The effect of H in y= ax(x)(x-h)+ k

In quadratics it is important to know where the vertex is. But we are not always given the vertex. It may be given to us in this form : y= 2(x-4)(x)+k. When it is given to us like this, we know that the step pattern is being multiplied by 2 and stretch by 2 which makes it steeper. and we are give "h". h represents the translation on the x-axis. but its kind of tricky. when we are give a positive 12, it means the parabola is moved negative 12 units on the x-axis. but when it is a negative 12, it is a positive 12 translation on the x-axis. This is horizontal translation. See example below.
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The effect of K in y=ax(x)(x-h)+k

The K in this formula represents how the parabola is going to move vertically. It is the same concept of the representation of H, except, instead of using the opposite sign, it all stay the same. you just have to move the vertex to whatever k represents. this is vertical translation. see example below.
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