Quadratic Graphs

What are Quadratic Graphs?

In figure 1 you can see two quadratic graphs. As you can see in the labels, the highest power in the polynomial is that of \(x^2\). Read on more to find out why this is.

In order to understand quadratic graphs, we'll introduce the Standard form of a quadratic function.

Standard form of a quadratic function

Tweaking any of these constants will have a visible effect on the graph of the function. The reason for which this version is utilized in graphing over the general form \[f(x)=ax^2+bx+c\] is that, although the constant \(a\) has the same effect in both forms,

• in standard form, shifting any of the constants \(h\) or \(k\) has a one-dimensional effect, i.e., the graph moves to the right or to the left, up or down (as you'll see further on the article);

• whereas shifting the constant \(b\) in the general form has a two-dimensional effect, i.e., the graph moves up and right or down and left, thus complicating manually graphing this form.

Now that you know more about the standard form of quadratic functions, let's move to see how this form can help us graph quadratics.

Graphing quadratics in standard form

Before going through the graphs of quadratic functions in standard form, you'll learn more about crucial concepts that are essential to graph our quadratics perfectly.

Let's start with the Opening of a Parabola.

Opening of a parabola

With reference to the standard form of a quadratic function, we will now explore the effects of varying each of the constants, \(a\), \(h\) and \(k\), individually.

The constant \(a\) has two effects on the parabola of a quadratic function, the first of which is dilation, that is, expanding or contracting the parabola according to its absolute value. You'll see more on this in the section Dilation.

The second effect is on defining the direction of the opening of the parabola according to its sign:

• If \(a\) is positive, \(a>0\), the parabola opens upward;

• If \(a\) is negative, \(a>0\), the parabola opens downward.

The effects that the sign of \(a\) has on the opening of a parabola can be seen in figure 2.

Fig. 2 - Effects of the sign of \(a\). Left: positive \(a\) opens upward. Right: Negative \(a\) opens downward.

Notice that a value of \(a\) of \(0\) results in a straight, horizontal line:

\[\begin{align} f(x)&=0(x-h)^2+k \\ \\ f(x)&=k \end{align}\]

Now, that you have learned about the effect of the value of \(a\) on the opening of a parabola, let's move to Translation.

Translation

Let's now have a look at the translation effects of constants \(h\) and \(k\) on parabolas.

First, let's introduce an important component of parabolas. The knowledge of this component will help you better monitor translations effected by those constants.

This point is also called turning point.

Here's a quick example to familiarize yourself with this term.

We can now move to the effect of \(h\) and \(k\) on quadratic graphs, or parabolas.

Let's start by looking at how the value of \(h\), in the term \((x-h)\) of the standard form, represents a horizontal translation (or shift) of a parabola.

• A positive \(h\) value, \(h>0\) will shift the graph \(h\) units to the right from the origin;

• A negative \(h\) value, \(h<0\) will shift the graph \(h\) units to the left from the origin.

Let's examine figure 3 to see how this would work.

Now, similar to how \(h\) represents a horizontal translation, \(k\) represents a vertical translation.

• A positive \(k\), \(k>0\), will shift the graph \(k\) units up from the origin;

• A negative \(k\), \(k<0\), will shift the graph \(k\) units down from the origin.

This principle is exemplified in figure 4.

Now that we have learned more about the effect of the value of \(h\) and \(k\) on the horizontal and vertical shift of the parabola from the origin, we are now ready to explore the last concept of Dilation.

Dilation

Let's now have a look at the dilation effect \(a\) can have on parabolas.

• An absolute value of \(a\) greater than \(1\), \(|a|>1\), contracts a parabola, moving it closer to the \(y-\)axis;

• An absolute value of \(a\) lower than \(1\), \(0<|a|<1\), dilates (or expands) a parabola, moving it away from the \(y-\)axis.

Once again, let's examine this transformation with figure 5.

Having grasped the transformations that the constants \(a\), \(h\) and \(k\) have on quadratic graphs, you're now ready to plot a quadratic graph from scratch!

Graphing Quadratic Functions

We will summarize the graphing procedure using the steps below.

Step 1. Finding the coordinates of the vertex.

To go from an equation in standard form to a graph, you can begin by finding the coordinates of the vertex.

Has you've seen before, since translations move the whole graph from its starting position, you can use the vertex as a point of reference to monitor the shift of the whole graph. And this is because the shape of the graph remains the same.

Hence, for a function \[f(x)=a(x-h)^2+k,\] the \(x\)-coordinate of the vertex is \(h\) and the \(y\)-coordinate is \(k\).

Thus, the coordinates of the vertex are \((h,k)\).

Step 2. Determine the opening of the graph, and if there's dilation (or contraction).

The next step is to determine whether the graph opens upward or downward. This is determined by the \(a\) value:

• If \(a>0\), the graph opens upward;

• If \(a<0\), the graph opens downward.

Also, examine if there is dilation or contraction:

• An absolute value of \(a\) greater than \(1\), \(|a|>1\), the graph contracts;

• An absolute value of \(a\) lower than \(1\), \(0<|a|<1\), the graph dilates (or expands).

Step 3. Determine the \(x-\) and \(y-\)intercepts.

The final step before sketching the graph, is to determine the \(x-\) and \(y-\)intercepts.

• The \(x-\)intercepts are determined by setting \(y\), or \(f(x)\), to zero and solving for \(x\), i.e., \[y=0 \text{ or } f(x)=0.\] The solution to this equation (or solutions, if there are any) will be the \(x\)-coordinates of the \(x\)-intercepts;

• The \(y-\)intercept can be found by setting \(x\) to zero and solving for \(y\), i.e., \[f(0)=a(0-h)^2+k.\] The answer will be the \(y\)-coordinate of the \(y\)-intercept.

Finally, when sketching, ensure you go through all the graph's intercepts and the vertex. Label these coordinates on your graph.

Now, we move to the practical part of this explanation, let's consider an example!

Quadratic Graph Examples

How to sketch quadratic equations?

We first deal with an example showing us how to sketch graphs of quadratic equations.

You may ask yourself now, do quadratic inequalities exist? and if the answer is yes, how do we graph them?

Well, quadratic inequalities are defined as all inequalities are defined, with the signs of inequality \(>, \ge, <, \le\).

In the next section, we shall learn more about them and their graphs.

Graphing Quadratic Inequalities

These have the same quadratic expressions as parabolas, except they have an inequality sign instead of an equal sign. This means that translations and contractions/dilations work the same as with the quadratic functions above.

There are some elements to learn which are unique to inequalities.

The first one is regarding the line of the graph, which can be dotted or continuous, depending on the direction of the sign of inequality:

• for \(\ge\) and \(\le\) inequalities, the line of the parabola will be continuous;

• for \(>\)and \(<\) inequalities, the line will be dotted.

Next, you must determine whether the shaded area (where the inequality is true) will be above or below the function. This, once again, depends on the direction of the inequality sign:

• for \(>\) and \(\ge\) signs, the shaded area will be above the curve traced by the parabola;

• for \(<\) and \(\le\) signs, the area will be below the curve.

This information can be summarized in the table below.

The following graphs show some examples.

In figure 7 we see the area and lines for the inequalities \(y >x^2\) and \(y\ge x^2\).

Figure 8 shows the area and lines for the inequalities \(y<x^2\) and \(y\le x^2\).

Fig. 8 - Graphic showing conditions where left: \(y <x^2\) and right: \(y\le x^2\).

The images contain graphs where the shaded area represents where the equation holds true.

Although these graphs do not have any contractions or vertical and/or horizontal translations, these would work exactly the same as with any quadratic graphs.

Now, we move to an example of graphing quadratic inequalities.

How to sketch quadratic inequalities?

Now, we deal with an example showing us how to sketch graphs of quadratic inequalities.

How to Determine the Function of a Quadratic Graph?

To go from a graph to an equation in standard form, you will need the vertex and any other point on the curve.

• The first step is to find the \(h\) and \(k\) values from the vertex, by simply substituting these coordinates into the vertex equation.

• Then, the value for \(a\) can be solved by substituting any other known point across the curve.

If the function is an inequality, the sign can be determined by the placement of the shaded area and the type of line.