Objective: We will first look at each standard form of the graph of a parabola. We will examine the parameters of each equation. We will then show a deviation from one equation to the other. Lastly, we will predict a sketch of several functions based on given parameters.
Standard Form
The first standard form of the parabola is as follows.
We will explore the parameters a, b, and c. The most common parabola that students have seen is when a=1, b=0, and c=0. This function is displayed below.
First we will explore what changing the parameter a does to the parabola. Let us watch as a varies from -5 to 5.
From this animation we can draw several conclusions.
If a>0, the parabola opens upwards and if a<0, the parabola opens downward.
If | a | > 1, the parabola narrows and if | a | < 1, the parabola widens.
Next we will explore what changing the parameter b does to the parabola. Let us watch as b varies from -5 to 5.
At a first glance, we can see that changing b will change the vertex of the parabola. The vertex is defined as the lowest point for parabolas that open upwards or the highest point when parabolas open downwards. The x-coordinate of the vertex is given by x=-b/(2a). For the positive values of b, the parabola is shifted down and to the left. For the negative values of b, the parabola is shifted down and to the right. We can look at this trend in the several examples below.
Last we will explore what changing the parameter c does to the parabola. Let us watch as c varies from -5 to 5.
We can see that changing the value of c moves the parabola in the vertical direction. Positive values of c result in a vertical shift upwards and negative values of c result in a vertical shift downwards.
Vertex Form
The next form of the parabola (vertex form) is as follows.
The most common parabola that students have seen is when a=1, h=0, and k=0. This function is displayed below. When we change a, the same changes occur that we explored when looking at the standard form of the equation. Thus we will only explore h and k.
First we will explore what changing the parameter h does to the parabola. Let us watch as h varies from -5 to 5.
We can see that changing the value of h causes a horizontal shift of the parabola. First note that the vertex form of the equation has a minus sign before the h value. When we have positive h values, the function shifts to the right and then we have negative values of h, the function shifts to the left.
Next we will explore what changing the parameter k does to the parabola. Let us watch as k varies from -5 to 5.
We can see that changing the value of k causes a vertical shift of the parabola. When k is positive, the graph shifts upwards and when k is negative, the graph shifts downwards.
An important discovery is that the vertex is given by the point (h,k). Thus, a benefit to the vertex-form of the equation is that it is east to plot the vertex by simply pulling out the h and k value.
Deviation from Standard Form to the Vertex Form
By completing the square, we can deviate from one form of the equation of the parabola to the other. We will show the deviation starting with the standard form of the equation.
From the deviation we can explicitly see what the vertex is in the standard form.
The x-value of the vertex is given by
and the y-value of the vertex is given by
.
Predictions
1.
First we will examine our parameter a. Since a is positive and | a | > 1, our parabola opens upwards and is more narrow than the standard parabola. Next we will examine b, which helps us find the x-coordinate of the vertex. When we compute -b/(2a) we have that our x-coordinate of the vertex is x=2/6. The c parameter of 4 is a vertical shift upwards. Below is the graph of our function with the standard parabola outlined in red.
2.
First we will examine our parameter a. Since a is negative and | a | < 1, our parabola opens downwards and is wider than the standard parabola. Next we will examine b, which helps us find the x-coordinate of the vertex. When we compute -b/(2a) we have that our x-coordinate of the vertex is x=5. The c parameter of -2 is a vertical shift of the standard vertex downwards. Below is the graph of our function with the standard parabola outlined in red.
3.
First we can find the vertex of our parabola as (3,-2). Then our value of a is positive and | a | > 1. Thus, our parabola opens upwards and is more narrow than the standard parabola.
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