Want to understand parabolas without having to read a textbook? This post demystifies parabolas by describing their most important features. Knowing these points will make any discussion on these graphs easier to follow.
Functions of the form $$y = ax^2 + bx + c$$ are known as a quadratic functions. Graphs of quadratic functions are known as parabolas.
Roughly speaking, a parabola resembles an elongated bowl. Three questions can be asked.
- Where is the ”bottom” of this bowl?
- Is the bowl right side up or upside down?
- How wide is the bowl?
It turns out that knowing the answers to there three questions gives everything you need to know about a parabola.
Let’s start with the quadratic function $$ y = x^2 $$ This is plotted below.
The ”bottom” of this parabola (called the vertex) is indicated by the red dot in the above graph, which is \((0,0)\).
Let’s say I want to change this parabola. First, I can move it. How? Let’s say I want to move the “bottom” of the parabola from \((0,0)\) to \((2,1)\). It turns out that if $$ y = (x \, – s)^2 + t $$ then the resulting parabola will look like the previous one, but the vertex (the “bottom”) will be \((s, t)\).
Since I want to move the vertex from \((0,0)\) to \((2,1)\), I should set \(s = 2\) and \(t = 1\) in the above formula to get \( y = (x \, – 2)^2 + 1\).
I get the following parabola as a result.
Let’s say I want to “flip” the “bowl” by “flipping” the graph of \( y = (x \, – 2)^2 + 1 \). It turns out that the graph of $$ y = -(x \, – s)^2 + t $$ is the same as the graph of $$ y = (x \, – s)^2 + t $$ except that the “bowl” is “flipped”.
The graph of \( y = -(x \, – 2)^2 + 1\) is as follows. Note that the vertex of the parabola does not change.
The above graph is a graph of the equation \( y = -(x \, – 2)^2 + 1 \). This can be written as \( y = a(x \, – 2)^2 + 1 \) with \( a = -1 \)
Let’s say I changed \( a \) to the negative number \( -10 \). Because \( -10 < -1 \), the resulting graph will be narrower than before. Specifically, the graph of \( y = -10(x \, – 2)^2 + 1\) is as follows. Note that the vertex does not change.
