In this explainer, we will learn how to graph any quadratic function that is given in its standard, vertex, or factored form using the key features of the graph of the function.
Recall that a quadratic function is a polynomial function where the highest order of one of the variables is 2. It also takes the form in the definition given below.
Definition: Quadratic Function
A quadratic function is a function that can be described as where , , and and .
We call the graph of a quadratic function a parabola, as this is the shape it takes.
Depending on whether in is positive or negative, the parabola opens upward or downward, as seen in the figures below.
There are different approaches to graphing quadratic functions. The first is using a table of values. This means when choosing or when given values for , we then generate values for by substituting them into the quadratic function we are trying to graph. Once we have done this, we can use the ordered pairs to plot the coordinates on a set of axes and then sketch the graph of the quadratic function using these coordinates and our knowledge of the shape of the quadratic function.
We will explore how to sketch the graph of a quadratic function in our first example.
Example 1: Finding the Graph of a Quadratic Function Using a Table of Values
Which of the following graphs represents the equation ?
Answer
To find the graph that represents the function we can use a table of values. To do so, we need to choose some values for that we are going to substitute in order to find their corresponding -values that then make up coordinates we can use to sketch the graph.
As a quadratic function is a parabola, it is helpful to use at least 5 coordinates to start with to sketch the graph, and then more if it is unclear what its shape is. It does not matter which -values you generally choose, but values that are close to zero are easier to calculate. Therefore, we will choose , , 0, 1, and 2 as our -values to begin with, as seen in the table of values below.
| 0 | 1 | 2 | |||
To calculate the -values, we substitute each -value above into the function . Doing so, we get the following.
When ,
When ,
When ,
When ,
When ,
Having calculated the values of , we can now put them into our table of values.
| 0 | 1 | 2 | |||
| 7 | 4 | 3 | 4 | 7 |
This means that our function passes through the points , , , , and . If we plot them on a set of axes, we get the following.
Looking at the graph, we can see it takes the shape of an upward-facing parabola, which is what we would expect as the coefficient of the -term is positive. It also appears that the turning point of the quadratic function is at the point . If we sketch the graph of the function going through the points, we get the following.
If we compare this graph with the options given, then we can see that only option E has the same turning point as our graph. We can also see that the graph in option E passes through the same points as our graph; therefore, the answer must be option E.
We can see from the previous example that using a table of values is one method of finding the graph of a quadratic function. However, we can also find the key features of the graph and use these to sketch the graph of functions.
In the first example, the function was given in standard form. This is when a quadratic function is written in the form
When a function is written in this form we can easily find the -intercept from the function. We can also find the turning point, called the vertex, and the -intercepts in order to graph the function.
First, we will discuss how to find the -coordinate of a quadratic function in standard form. Now, to find the -intercept of any function, we use the fact that the -coordinate is zero and substitute into the function. Doing so with a quadratic function in standard form gives us
Therefore, the -intercept of a quadratic function in standard form is , the constant term.
Second, we will discuss how to find the -intercepts of a quadratic function in standard form. Similar to before, to find the -intercept of any function, we use the fact that the -coordinate is zero and substitute into the function. Doing so with a quadratic function in standard form gives us
Recall that to solve a quadratic equation equal to zero, we have three key algebraic methods: factoring, completing the square, and the quadratic formula. As some quadratics are not factorable, then we can use either completing the square or the quadratic formula. We will generally use the quadratic formula in cases where the function is in standard form. Letβs recall the quadratic formula.
Definition: The Quadratic Formula
To solve a quadratic equation in the form , , with variable and constants , , and , we can use the quadratic formula to solve for , when is greater or equal to zero.
Therefore, using the formula above, when they exist, the -intercepts of a quadratic function in standard form are and .
Third, we will discuss how to find the vertex of a quadratic function in standard form. Since quadratic functions are symmetrical about the vertex, then the line of symmetry passes vertically through the vertex. This also means that the line of symmetry passes through the midpoint of the -intercepts. Using this fact we can derive a formula to find the -coordinate for the vertex.
Since the -coordinate of the vertex is the midpoint of the -intercepts, then we get the following from the midpoint formula:
Now, since and cancel out, we get
Therefore, to find the -coordinate of the vertex for a quadratic function in standard form, we use the formula
To find the -coordinate of the vertex, we substitute the -coordinate back into the quadratic function. We can use the same formula for the vertex, even in the case when we do not have two -intercepts.
Once we have found the -coordinate, -coordinates, and vertex, we can use these features to sketch the graph of the quadratic function. We can also check whether the coefficient of the -term is positive or negative to determine whether the parabola opens upward or downward, respectively, to help us sketch the graph. Letβs summarize this in the guide below.
How To: Finding the Key Features of the Graph of a Quadratic Function in Standard Form
For a quadratic function in standard form , , we can sketch its graph by finding its key features and the direction in which it opens by using the following steps.
Step 1: Finding the π-Intercept
The -intercept is .
Step 2: Finding the π-Intercepts
When they exist, the -intercepts are and , which are found using the quadratic formula (or otherwise).
Step 3: Finding the Vertex
The -coordinate of the vertex is , and the -coordinate is found by substituting the -coordinate into the function.
Step 4: Determining Whether the Graph Opens Upward or Downward
The graph opens upward if and downward if .
Next, we will find the graph of a quadratic function in standard form using the steps we discussed above.
Example 2: Finding the Graph of a Quadratic Function in Standard Form
Which of the following graphs represents the equation ?
Answer
To find the graph of the function , we can find its key features first and then use these to graph it.
We can see that the quadratic function is in standard form, or in the form
When a quadratic function is in this form, we use certain techniques to find the -intercept, -intercept, and vertex of the graph.
We know that the -intercept is when the -coordinate is zero. Therefore, we get a -coordinate of , which in the case of is 6. So, the -intercept is .
Similarly, we know that the -intercept is when the -coordinate is zero. This gives us which is a quadratic equation. Recall that to solve a quadratic equation, we can use factoring, completing the square, or the quadratic formula. As this does not appear to be easily factorable, and since it is not as straightforward to complete the square without needing fractions, we will use the quadratic formula.
The quadratic formula states that to solve an equation in the form , , then we have
For the equation , where , , and , this gives us
Therefore, the -intercepts are and , correct to 2 decimal places.
To find the -coordinate of the vertex in standard form, we can find the midpoint of the -intercepts, which gives us
To find the -coordinate of the vertex, we substitute the -coordinate, , into the function
Therefore, the coordinates of the vertex are .
Now we have found the intercepts and vertex, we can sketch the graph of the function. Plotting this information on a set of axes, we get the following.
Using this information, we can sketch the graph. Further, we can also use the coefficient of the -term, which for is negative, to tell us that the function opens downward. Sketching the graph then gives us the following.
Comparing this with the different options, we can see that only options D and E open downward, so it must be one of these options. We can see from the vertex that option D has the same vertex as our graph. Therefore, the answer must be option D.
Having covered how to sketch a graph where the function is in standard form, we will now discuss how to sketch a graph where the function is in vertex form.
By vertex form, we mean the form that allows us to easily read the coordinates of the vertex from the function. This is where a function is in the form
When a function is in this form then the coordinates of the vertex are .
To sketch a graph in vertex form, it is helpful to find the key features of the graph, the -intercepts, -intercept, and the vertex (just as we did for the standard form).
As before, to find the coordinates of the -intercept, we set and solve for . Similarly, to find the coordinates of the -intercepts, we set and solve for . As the vertex form is similar to that we get when completing the square, then this is usually the best method to use to solve the quadratic equation.
To decipher whether the graph opens upward or downward, we use as before, and whether it is positive or negative, to determine if the graph opens upward or downward respectively.
With all of the key features found, we can then use these to graph the quadratic function in vertex form. This is summarized in the guide below.
How To: Finding the Key Features of the Graph of a Quadratic Function in Vertex Form
For a quadratic function in vertex form , , we can sketch its graph by finding its key features and the direction in which it opens by using the following steps.
Step 1: Finding the Vertex
The vertex is .
Step 2: Finding the π-Intercept
The -intercept is when . To find it, substitute into the function and solve for .
Step 3: Finding the π-Intercepts
When they exist, the -intercepts are when . To find them, substitute into the function and solve for by completing the square, or otherwise.
Step 4: Determining Whether the Graph Opens Upward or Downward
The graph opens upward if and downward if .
In the next example, we will explore how to find the graph of a quadratic function in vertex form by finding the features of the graph.
Example 3: Finding the Graph of a Quadratic Function in Vertex Form
Which of the following graphs represents the equation ?
Answer
We are asked to find the graph of the function . We can see that the quadratic function is in vertex form, which is generally
To sketch a function in vertex form, we can use the features of the graph to help us.
The first feature we can find, as the name suggests, is the vertex. Generally, this is for . Therefore, for the function , the vertex is at . Note that at this point only two of our options have the same vertex, and we can deduce which option it is by looking at whether is positive or negative, but we will consider how to graph the function as if we could not decipher this.
Second, we can find the -intercept. For all functions, the -intercept is when , so we can find the value of by substituting into . This gives us
Therefore, the -intercept is .
Third, we can find the -intercepts. Similar to before, we know that for all functions, the -intercepts are when , so we can find the -intercepts by substituting into the function and solving for . This gives us
As this is a quadratic equation, we need to use either factoring, completing the square, or the quadratic formula to solve this. As it is already partially solved using the method for completing the square, we tend to use this method for solving equations in vertex form. Doing so, gives us
Therefore, the -intercepts are and .
Having found the vertex, the -intercept, and the -intercepts, we can now plot these on some axes in order to help graph the function, as seen below.
Using this information, we can sketch the graph. Further, we can use whether is positive or negative to determine whether the graph opens upward or downward. Since , which is positive, then the graph open upward. Sketching the graph then gives us the following.
From the options given, we can see that options B, C, and D open upward, so our graph must be one of these. We can see that option D has the same -intercepts and vertex; therefore, the answer must be option D.
Having covered how to sketch a graph from a function in standard and vertex form, we will now discuss how to sketch a graph of a function in factored form. Factored form is when a quadratic function is written as a product of its factors, or
As with the other forms, we can find the key features of the graph to help us graph the function.
In this form, it is easy to decipher the -intercepts of the function. We can determine the -intercepts by letting and solving for . This gives us and since one of the factors must be zero, we have
So, the -intercepts are therefore and .
To find the -intercepts, we simply substitute into the function and solve for .
To find the coordinates of the vertex, we use the symmetry of the graph about the vertex; we know that the vertex is the midpoint of the -intercepts, meaning that the -coordinate of the vertex is halfway between the -intercepts. Using the midpoint formula gives us
Therefore, the -coordinate of the vertex is . To find the -coordinate, we substitute this back into the function and solve for .
To decipher whether the graph opens upward or downward, we use as before, and whether it is positive or negative, to determine if the graph opens upward or downward respectively.
Letβs summarize these points into the guide below.
How To: Finding the Features of a Graph of a Quadratic Function in Factored Form
For a quadratic function in factored form , , we can sketch its graph by finding its key features and the direction in which it opens by using the following steps.
Step 1: Finding the π-Intercepts
The -intercepts are and .
Step 2: Finding the π-Intercept
The -intercept is when . To find it, substitute into the function and solve for .
Step 3: Finding the Vertex
The -coordinate of the vertex is . To find the -coordinate, substitute it into the function and solve for .
Step 4: Determining Whether the Graph Opens Upward or Downward
The graph opens upward if and downward if .
In the next example, we will explore how to find the graph of a quadratic function in factored form by finding the features of the graph.
Example 4: Finding the Graph of a Quadratic Function in Factored Form
Which of the following graphs represents the equation ?
Answer
To find the graph of the quadratic function , we can find the features of the graph and use them to sketch the graph. We will find the intercepts and vertex to graph the function.
As the function is in factored form, as in then it is easy to first find its -intercepts, as these are and .
So, for the quadratic function , the -intercepts are and . Note that at this point we can deduce which of the options is correct, but for completeness, we will consider how to graph the function using all the features.
Next, we can find the coordinates of the -intercept by substituting . Doing so gives us
Therefore, the coordinates of the -intercept are .
To find the coordinates of the vertex, we can use the symmetry of the graph about the vertex. Since the -coordinate of the vertex is halfway between the -intercepts, then generally for a function in factored form, the -coordinate of the vertex is
For our function, and , so the -coordinate of the vertex is
We then take the -coordinate of the vertex and substitute it back into the function in order to find the -coordinate. Doing so gives us
Therefore, the coordinates of the vertex are .
Now we have found the -intercepts, -intercept, and vertex, we can plot these coordinates onto a set of axes to help us graph the function. Doing so gives us the following.
Using this information, we can sketch the graph. Further, we can use whether is positive or negative to determine whether the graph opens upward or downward. Since , which is positive, then the graph opens upward. Sketching the graph then gives us the following.
From the options given, we can see that only option E has the same -intercepts, -intercept, and vertex; therefore, the answer must be option E.
So far, we have considered how to find the graph of a function given its equation. Next, we will discuss how to find the equation of a function given its graph. Generally, we consider what features of the graph are given and what form we want our equation to be in to help us determine the approach.
In the last example, we will discuss how to find the equation of a function in vertex form given its graph.
Example 5: Finding the Equation of a Function in Vertex Form given its Graph
Which of the following is the equation of the function drawn on the graph?
Answer
To determine which function matches the graph given, it is helpful to use the key features of the graph. As the functions given are all in vertex form, which is then we know that for a function in this form the coordinates of the vertex are .
From the graph, we can see that the coordinates of the vertex are , as indicated by the point on the graph below.
This means that and , giving us the equation
To find , we need to substitute a point on the graph into the equation. It is usually easiest to use one of the intercepts to do this.
We can see from the graph that it is not clear what the exact values of the -intercepts are, but we can determine the exact value of the -intercept. Therefore, we will use this point.
Looking at the graph, we can see that the -intercept is , which is shown by the point on the graph below.
This means that when , then . Substituting this into and solving for , we get
We can see that as the graph opens downward, then must be negative, which is the case here.
Substituting into , we can then determine the function of the graph, which is
This means that the answer is option B.
In this explainer we have learned how to graph quadratic functions using different approaches when they are in different forms. We have also learned how to find the equation of a function from its graph. Letβs recap the key points.
Key Points
- One method of graphing a quadratic function is to use a table of values. We do so by choosing at least 5 points for and substituting these into the function to find . We can then use the points to sketch the graph.
- For a quadratic function in standard form, ,
, we use the following features to help graph the
function:
- the -intercept ,
- the -intercepts by substituting and using the quadratic formula, or, otherwise, to find the -coordinates, or whether they exist,
- the vertex using the formula for the -coordinate and then substituting to find the -coordinate,
- whether is positive or negative to determine if the graph opens upward or downward.
- For a quadratic function in vertex form,
, , we use the
following features to help graph the function:
- the vertex ,
- the -intercept by substituting and solving for ,
- the -intercepts by substituting and using completing the square, or, otherwise, to find the -coordinates, or whether they exist,
- whether is positive or negative to determine if the graph opens upward or downward.
- For a quadratic function in factored form, ,
, we use the following features to help graph the function:
- the -intercepts and ,
- the -intercept by substituting and solving for ,
- the vertex using the formula for the -coordinate and then substituting to find the -coordinate,
- whether is positive or negative to determine if the graph opens upward or downward.
- To find the equation of a quadratic function from its graph, we identify its key features and use these to find the equation in the form required.