In this explainer, we will learn how to solve quadratic equations using function graphs.
Let us recall the definition of a quadratic equation.
Definition: Quadratic Equation
A quadratic equation is an equation that can be written in standard form as where is the variable, , , and are constants, and .
We note that it is always possible to rearrange a quadratic equation to be equal to zero, as shown above, by taking all of the variables and the constant to one side of the equation.
Now, recall that when we solve a quadratic equation, we find the values of for which the equation is satisfied. One of the ways we can solve a quadratic equation is by factoring. This means we rearrange the quadratic equation into the factored form:
From here, it is possible to deduce that both and satisfy the equation and that they are, therefore, solutions of the equation. In this explainer, we will see how we can also use a graphical method to solve a quadratic equation. To plot a quadratic equation, we rewrite it as a function:
In other words, we replace 0 with the variable . Note that we will often denote the left-hand side of the function by (i.e., ). Writing our equation as a function allows us to show graphically how changes for different values of . We do this by drawing the curve that makes on a graph with - and -axes.
Suppose we then want to solve the quadratic equation using this graph. Since a quadratic equation is solved when it is equal to 0, we set in the function and find the values of for which the equation is satisfied. Thus, the solutions of the equation are the values for which the function is zero, which we refer to as the roots of the function. On a graph, these values are the -coordinates of the points where the -value is zero, which corresponds to the points at which the graph crosses the .
Now, graphs of quadratic functions have distinctive properties which can be used to help us identify points of interest about an equation. Whether inspecting the graph of a quadratic function or using an equation to draw the graph, the following points are important to remember.
Properties: Graphs of Quadratic Functions
- The graphs of quadratic functions of the form have distinctive parabola shapes,
as shown below.
- They have a minimum vertex with the curve opening upward when the value of is greater than zero (as
shown in the left graph). They will have a maximum vertex with the curve opening downward when is less
than zero (as shown in the right graph).
Note that the value of cannot be equal to zero, as this would mean that there is no term and, as such, the corresponding equation would not be quadratic. - A quadratic function can also be rearranged into vertex form as where are the coordinates of the vertex of the parabola (i.e., the turning point).
- The graph of a quadratic function is symmetrical about the vertical line , where is the vertex of the parabola.
- The -intercept (the coordinate where the parabola crosses the ) of the function will always be at .
- The -intercepts, where the curve crosses the , will be the points at which . The -coordinates of these points are the roots of the function and correspond to the solutions of the original quadratic equation. In a graph, we can identify these points via inspection.
- It is useful to remember that a quadratic equation will have up to two real solutions. If an equation has two solutions, the
corresponding function will have a graph that crosses the twice. An equation with a repeated solution
will lead to a graph that has a vertex on the . Finally, an equation having no solution will mean that
the graph is entirely above or below the .
- In the graphs above, the first function has two roots, the middle function has one root where the graph touches the , and the last function has no roots.
Let us now look at an example where we can apply the above properties to find the solution to a quadratic equation using a graph.
Example 1: Solving a Quadratic Equation Graphically
The diagram shows the graph of . What is the solution set of the equation ?
Answer
Let us recall that the coordinates of any point on the graph of a function are given by . We are being asked to find the solution set of the equation , which is the set of values of for which the -value is 0. On the graph, this corresponds to the points at which the curve crosses the , since at these points. By inspecting the curve, we can see that it crosses the at two points: when and when .
Therefore, the solution set is .
In the first example, we saw that, since the curve crossed the twice, the equation had two solutions. Let us consider an example where that is not necessarily the case.
Example 2: Solving a Quadratic Equation Graphically
The diagram shows the graph of . What is the solution set of the equation ?
Answer
Let us recall that the coordinates of any point on the graph of a function are given by . Thus, to find the solution set of , we must find the points for which and find the corresponding values. We can find these on the graph by identifying the points where the curve crosses the . In this situation, we can see that the curve does not actually cross the , but it does touch it at one point, when . Hence, we can say that is equal to 0 at the point where .
Thus, the solution set has just one value, .
So far we have seen graphs of quadratic equations that cross the twice or just touch it once, but what happens when the curve does not intersect the at all?
Example 3: Solving a Quadratic Equation Graphically
The graph shows the function . What is the solution set of ?
Answer
Here, we have been given the explicit function that describes , but, since we have the graph, we can simply solve the equation graphically without factoring or using the quadratic equation. Recall that the solution set of can be found by identifying the points on the graph where , which is where the curve intersects the . In this case, however, the curve is entirely above the . For this reason, there are no points at which .
Hence, there are no real values of that solve the equation, so the solution set is .
We will now look at an example of how we can draw our own graph to solve an equation. Here, we will use a table of values to find a number of coordinates to plot and then join them to make a curve. It is useful to remember that this approach allows us to plot the graph of any polynomial function corresponding to an equation.
Example 4: Plotting the Graph of a Quadratic Function and Using It to Solve an Equation
By drawing a graph of the function , find the solution set of .
Answer
We can begin by drawing a graph of the function by finding coordinates on the curve. We can do this by setting up a table of values and calculating for the selected -values.
0 | 1 | 2 | 3 | |||
---|---|---|---|---|---|---|
8 | 2 | 0 | 2 | 8 | 18 | |
0 | 3 | 6 | 9 | |||
14 | 5 | 0 | 2 | 9 |
Taking the values from the rows and , we get the points , , , , , and . Let us plot these on the -plane and join them with a smooth curve.
We now need to find the solution set of . Recall that the solutions of the equation are the -values of the points where the graph of the function crosses the . We can see that the graph crosses the at two points: the first is the point labeled , which is at , and the second seems to be at . We can verify this is the correct value by putting it into the formula for . This gives us
This shows us that is indeed a solution of the equation. Thus, the solution set is .
Just as it is possible for us to draw a graph to help us solve a quadratic equation, it is also possible for us to solve a quadratic equation to help us determine the correct graph of the corresponding function. In the following example, we will consider how to factor an equation and use the answer to figure out what the graph must look like.
Example 5: Recognizing the Graph of a Quadratic Function after Solving It via Factoring
Solve by factoring, and hence determine which of the following figures would be a sketch of .
Answer
Part 1
As we have been asked to solve by factoring, let us first remember how to do this. We want to factor the equation using two unknown values and as follows:
Matching the coefficients, we can see that this requires and . Since the product of and is negative, that means one of or is negative and the other is positive. Let us assume is negative. Thus, let us consider the four possible pairs of and that multiply to make :
1 | 6 | 2 | 3 |
Out of these options, only and give us . Thus, the correct factoring is:
Having factored the equation, we can solve it by setting it to 0 and finding what values of satisfy the equation. In other words, we have
This is satisfied when or , which leads to or .
Part 2
We now need to determine which of the given figures is a sketch of . Recall that the roots of a function (which are the same as the solutions of the corresponding equation) tell us what values of give us . This means we know at what points the curve will cross the : and . Considering the five graphs, only one crosses the at these points: option E.
Additionally, let us note other aspects of the graph that we can use to verify the correct answer. We note that the -intercept of the graph (the point where the curve crosses the ) is . Recall that, for a quadratic function of the form , the -intercept is . Since the function is , this gives us , which is indeed correct. Furthermore, we can see that, as (i.e., ), the graph must open upward, which is exactly what happens in graph E. This further confirms for us that E is the correct answer.
In conclusion, the answer is option E.
In some questions, we will be given a quadratic equation that has to be rearranged before we can solve it graphically. Here is an example of this type.
Example 6: Rearranging and Solving a Quadratic Equation Graphically
Find the solution set of the equation .
Answer
As we have been asked to find the solution set of the equation , we must first rearrange it into the form . Note that we can do this by subtracting and 10 from both sides to get
Recall that the solutions of the equation are the -values of the points where the graph of the function crosses the (since at these points). We can find these values by sketching the graph of the function and finding where it crosses the . We can draw the graph by setting up a table of values and calculating for the selected -values, as follows.
0 | 1 | 2 | 3 | 4 | 5 | 6 | ||||
8 | 0 | 0 | 8 |
Then, we plot these points on the -plane and join them with a smooth curve, which gives the graph below.
It is easily seen that the curve crosses the at and . Note that we could have read off these values directly from our table, which showed that for these two values of .
We conclude that the solution set of the equation is .
So far, we have exclusively dealt with quadratic equations of the form or and considered how these relate to functions of the form when . Equally, we can form other quadratic equations by choosing to be equal to different values. For instance, if we chose , we would have the equation which can be rearranged into
In other words, choosing results in another quadratic equation where the constant term is . Solving equations of this form graphically can be done in almost the same way as for : instead of considering where the curve crosses the (i.e., ), we consider where the curve crosses the line , where is a constant (it is 3 in our example above). This can be done by drawing a horizontal line on the graph at that value of . In our final example, we will consider an application of this idea.
Example 7: Solving Two Quadratic Equations that Differ by a Constant Graphically
The graph shows the function .
- What is the solution set of ?
- What is the solution set of ?
Answer
Part 1
The solution set of is the set of values of for which the curve crosses the (since at these points). Inspecting the graph, we can see that the curve crosses the at two points: and . Therefore, the solution set of is .
Part 2
Just as the solution set of is the set of values of for which , the solution set of is the set of values where . We can find this by drawing a horizontal line at and determining the coordinates of the points where the line and the curve intersect.
We can identify from the graph that the points of intersection are at and . Therefore, the solution set of is .
Note that we can use the above method to solve problems taken from a real-world context, as in the next example.
Example 8: Solving a Quadratic Equation From a Real-World Context Graphically
A ball is thrown vertically upward from the ground with velocity . The height of the ball above the ground, in metres, is given by the equation . The graph of this function is shown below.
Work out the times at which the ball is at a height of 49 m above the ground.
Answer
Recall that the solution set of is the set of values of for which the curve crosses the (since at these points); these two values of would correspond to the times at which the ball is on the ground at the start and finish of its flight.
Similarly, the times at which the ball is at a height of 49 m above the ground correspond to the solution set of , which is the set of values of where . We can find these values by drawing a horizontal line at and determining the coordinates of the points where the line and the curve intersect.
We can identify from the graph that the points of intersection are at and , so the solution set of is . We conclude that the ball is at a height of 49 m above the ground after 2 seconds and 5 seconds.
Let us finish by considering the main things we have learned in this explainer.
Key Points
- We can solve a quadratic equation in a number of ways, either using an algebraic method, such as factoring, or graphically, by inspecting the graph of the corresponding function. The benefit of an algebraic solution is that it will result in an exact answer. A graphical method is an approximate solution; therefore, it is often more difficult to use when there are noninteger root values. However, it can be an easier and faster method of solving a quadratic equation, and it allows us to visually compare a number of quadratic equations.
- To solve a quadratic equation graphically, we first draw the graph of the corresponding function by creating a table of
values. We can then inspect the graph to find the points where the curve crosses the as shown below.
- It is important to remember that we may have two, one, or no solutions to a quadratic equation depending on whether the
function crosses the twice, once, or never, as shown below.
- We can solve quadratic equations formed by setting , by drawing a horizontal line at the value , and by finding where it intersects the function.