Very well know as the Parabola is the quadratic function, I love how it relates to the quadratic number pattern and the quadratic equation; it is more of the latter in a diagram, how interesting is math.

This function is expressed by the equation y= a(x-p)^2+q or by the quadratic equation of y=ax^2 + bx+ c. completing the square comes in very handy converting the second equation to get to the first one. The first equation can be referred to as the turning point form and the other as the standard form.

The variables a, p, and q

Concave up, concave down; sad or happy face they said, that is the effect of letter a in both equations. When a is negative, the graph will increase from left to right and reach a turning point, then start to decrease upto infinity. In this case, the turning point is called the maximum, because it is the maximum y value that the function will reach. Remember that this happens at the axis of symmetry.

What then when a is positive; of cause, positive means happy, the graph has a very wide smile and will decrease from left to right until it reaches the minimum value and then starts to increase until we have no more paper to draw on. The minimum value is the turning point and the lowest y value we can get from the function

Now that we have talked a lot about the turning point, how do we get this point. The turning point equation makes it easy, hence it was named after the turning point; using this one we simply take the values of p and q, but we have to change the sign of p. if p is negative we will make it positive and vice versa; this happens when we take the value of p out of the equation to co-ordinate of turning point, and once again when we take it from the co-ordinate back to the equation. (-p,q) is the turning point co-ordinate. If we are using the standard form equation; we use the formula -b/2a which is equal to the x-value of the turning point, once we have the x-value we will substitute it in the equation to find the y-value.

Enough about a. so, p represents the horizontal shift; either left or right, we subtract p points to move to the right and we add p points to move to the left. This is most likely why we need to change the sing when dealing with the turning point using p as the x value of the turning point. I like how simple q is, unlike p it does not complicate things, we simply add to shift the graph upwards and subtract to shift it downwards. This very well describes our vertical shifts.

Points of intersection with the axis

We have done this for a very long time in math, let x equal to zero and find the y-intercepts, let y equal to zero and find the x intercepts. Algebra will come in handy when solving for the unknown variable. On the standard form equation, c is always the y-intercept; but this is not the case with q on the turning point equation.

Sketching

To sketch the graph

• draw a cartesian plane
• find the turning point
• find the intercepts
• connect the above co-ordinates

concave up if a is positive

concave down if it is negative

that’s it about the parabola, use your study guide to get familiar to the questions and practice answering them. The above characteristics will help you understand and remember how to deal with the parabola. See you on the next post dealing with the hyperbola.

Operating Disclaimer:

This blog post is not the first step towards learning the concepts of math but the coupling step to ease the use of your study guide and other study material. Read it over and over again to keep the concepts at the back of your head. Math is a subject of rules; know the rules, be able to duplicate them on a blank paper and go fetch your distinction