Turning Points

In the English language, the phrase "reached its turning point" means that something has encountered a significant change. The same concept can also be applied to Mathematics!

Let us recall the graph of a quadratic function. Do you remember the name of the U-shaped curve it produces? That's right! It's called a parabola.

Parabola example, StudySmarter Originals

Now, notice how the graph curves down and hits a point where it then curves up. We see that the parabola encounters a change in direction upon meeting this exact location. This is called the turning point.

In this lesson, we shall look at the idea of a turning point and introduce several methods we can use to determine the turning point of a given function. Throughout this topic, we will only focus on finding the turning point of a quadratic function of the form,

$y=a{x}^2+bx+c$.

The Meaning of a Turning Point

As introduced at the beginning of this article, the turning point of a graph is a point at which the curve deviates from its initial path. That is to say, if a car was driving up a hill and reached its peak, it will then have to drive down in order to get to the other side of the hill. Below is a formal definition of a turning point.

In some textbooks, a turning point may also be referred to as a critical point.

A Turning Point on a Graph

The turning point of a curve is represented by a pair of x and y-coordinates on the Cartesian plane and is denoted by the point, (x, y). Below is an example of what a turning point may look like when it is plotted on a graph.

The turning point on the Cartesian plane, StudySmarter Originals

Types of Turning Points

There are two types of turning points we need to consider when graphing quadratic functions:

• Minimum Point, and

• Maximum Point.

Each type has its own orientation of curvature and can be determined by looking at the value of the leading coefficient of the quadratic function, i.e. the coefficient of x². The table below describes the key features of both these cases along with their general plot.

Turning Point Formula

The turning point of a quadratic equation coincides with the vertex of its corresponding parabola. To determine the coordinates of its vertex, we shall make use of the Vertex Formula, or in this case, the Turning Point Formula. Given the standard form of a quadratic equation,

$y=a{x}^2+bx+c$

the vertex form of this quadratic equation is

$y=a{(x-h)}^2+k$,

where (h, k) is the vertex or turning point of the parabola. There are two ways in which we can determine the values of h and k.

Method 1

Evaluate $h=-\frac{b}{2a}$ and $k=-\frac{D}{4a}$ where D is the discriminant and $D=b^2-4ac$.

Method 2

Calculate $h=-\frac{b}{2a}$ and use this result to find k by substituting this value of h into y.

Note that this formula can only be applied to quadratic equations as in the form above. It is not applicable to finding turning points of higher degree polynomials and other complex functions. However, this is where the next section will be of relevance! Here, we shall look into other ways in which we can locate turning points of any type of equation.

The Math of Finding the Turning Point

For a given quadratic function, we can locate the turning point of the parabola using four methods, namely:

1. Line of symmetry

2. Factorisation

3. Completing the square

4. Differentiation

In this segment, we shall provide a detailed step-by-step process of each technique stated above along with several worked examples.

Line of Symmetry Method

We shall begin by observing a distinct trait of a parabola that allows us to determine its turning point. This property is known as its line of symmetry. Recall that the line of symmetry is a line segment that divides an object precisely in half.

The graph of any quadratic function has a vertical line of symmetry that passes through its turning point. That is to say, the turning point is situated exactly in the middle of the parabola. To identify the turning point of a given curve, we can use the following steps.

Step 1: Determine whether the graph is a minimum or maximum by observing the coefficient of x²;

Step 2: Identify the equation for the line of symmetry using the formula,

This is the x-coordinate of the turning point;

Step 3: Evaluate the y-coordinate by substituting the x-coordinate from Step 3 back into the given function.

Below are two worked examples applying this method.

Given that a vertical line of symmetry cuts the parabola into two equal halves and crosses its turning point, we can say that the position of its turning point is exactly at the midpoint between the (two) x-intercepts of a quadratic function. With that in mind, we can find the turning point of a quadratic equation with an approach involving factorisation. There are five steps to this method.

Step 1: Determine whether the graph is a minimum or maximum by observing the coefficient of x²;

Step 2: Factorise the given quadratic function and set it to zero;

Step 3: Identify the roots of the function;

Step 4: Locate the x-coordinate of the turning point by taking the midpoint of the pair of roots;

Step 5: Evaluate the y-coordinate by substituting the x-coordinate from Step 4 back into the given function.

Let us now look at some worked examples that demonstrate this technique.

Completing the Square Method

Next, we shall now move on to finding the turning point of a quadratic equation using a method called completing the square. When we complete the square of a given quadratic function, it will take the form,

$y=a{(x-h)}^2+k$

To recall the process of this method, refer to the topic: Completing the square. There are 3 steps to consider in this case.

Step 1: Determine whether the graph is a minimum or maximum by observing the coefficient of x²;

Step 2: Complete the square of the given quadratic function so that it looks like $y=a{(x-h)}^2+k$;

Step 3: Identify the x and y-coordinates of the turning point. The value of h and k are the x and y-coordinates respectively.

Now that we have established the steps above, let us observe two worked examples.

Differentiation Method

Let us now look at our final method of locating the turning point of a quadratic function. This time, we shall be using a technique widely used in calculus called differentiation. Recall that differentiation is a method used to determine the gradient of a curve at a specific point. A more detailed overview of this topic can be found here: Differentiation.

For a quadratic function of the form$y=a{x}^2+bx+c$, the gradient of the curve at its turning point equals zero. In other words, $\frac{dy}{dx}=0$.

As mentioned before, a turning point can either be a minimum or maximum. Let us look at each case in turn.

To locate the turning point using differentiation, simply calculate the first derivative of the given function and set it to zero. This will give you the x-coordinate of the turning point. The y-coordinate is found by substituting this found x-value into the initial quadratic function.

$\frac{dy}{dx}=0$ or $f\text{'}\left(x\right)=0$

The nature of the turning point is found by calculating the second derivative of the given function. Then,

1. if $\frac{d^{2}y}{d^{2}x}>0$ or $f\text{'}\text{'}\left(x\right)>0$ then the turning point of our function is a minimum;

2. if $\frac{d^{2}y}{d^{2}x}<0$ or $f\text{'}\text{'}\left(x\right)<0$ then the turning point of our function is a maximum;

3. if $\frac{d^{2}y}{d^{2}x}=0$ or $f\text{'}\text{'}\left(x\right)=0$ then the turning point of our function is an inflection point.

Note that the third clause will be further discussed on the topic of cubic functions and higher-order polynomials.

As we now have these facts covered, let us now observe two examples.

Turning Point Examples

To further enhance our skills in finding the turning point of a parabola, we shall end our discussion with two real-world examples involving the concept of a turning point.

Try it yourself: Attempt using the other methods for finding the turning points for these two questions above. Do they give you the same answer?