In mathematics, we learn about quadratic functions. It is a quadratic function with a graph of squares, and the graph has a curve.

In mathematics involving graphs, we first learn direct proportionality. A graph that makes an application of proportional is a linear function. Further development of linear functions is quadratic functions. Linear and quadratic functions are almost identical in concept; the only difference is whether $x^2$ is in the equation or not.

However, the shape of the graph is quite different in quadratic functions. There are also features of quadratic functions that must be remembered.

In addition, quadratic problems are often combined with linear functions. So, we will explain how to solve quadratic function problems.

The Equation of $y=ax^2$ Is a Quadratic Function

When we assign a number to $x$, the equation that clearly determines the value of $y$ is a function. In a linear function, for example, we have the following equation

  • $y=x+2$

In such an equation, if $x$ is doubled, the increase in $y$ is also doubled. Therefore, the values of $y$ and $x$ are proportional to each other.

In quadratic functions, on the other hand, we have the following equation, for example.

  • $y=x^2$

In this case, if the value of $x$ is doubled, the value of $y$ is 22 times (four times). In other words, the value of $y$ is proportional to $x^2$. As the value of $x$ increases, the value of $y$ increases by multiplication of the square.

In a quadratic equation, for example the following.

  • $y=5x^2$
  • $y=-3x^2$
  • $y=x^2$
  • $y=\displaystyle\frac{1}{2}x^2$

Thus, the number in front of $x^2$ changes depending on the expression. Therefore, the quadratic function is represented by the following formula

  • $y=ax^2$

The most basic expression in quadratic functions is the formula represented by $y=ax^2$.

Many of the Natural Phenomena Are Quadratic Functions

Why do we need to learn quadratic functions? It’s because many of the phenomena around us are quadratic functions.

For example, what happens to the ball when you place it on a slope and take your hand off the ball? Right after you take your hand off the ball, the speed of the ball is slow. But over time, the speed of the ball gradually increases. It’s not proportional that is always a constant speed, but a quadratic function that increases the speed over time.

Other than that, when a stationary car speeds up, the speed increases in quadratic function. Of course, we use quadratic functions not only for cars, but also for calculating the speed of a rocket launch. It is important to understand that quadratic functions are used in every situation.

Table of Quadratic Functions Is Proportional to $x^2$

What does a quadratic function look like in a table? Unlike direct proportionality, as mentioned earlier, quadratic functions are proportional to $x^2$.

For example, in equation $y=x^2$, the table looks like this.

If $x$ is doubled, $y$ is quadrupled. If $x$ is tripled, $y$ is multiplied by 9. If $x$ is quadrupled, $y$ is multiplied by 16.

Since the equation is $y=x^2$, as the value of $x$ increases, the value of $y$ is $x$ squared. It is important to note that the value of $y$ is an expression that increases by the square.

Quadratic Functions Become a Parabolic Graph

What does a quadratic formula look like when written on a graph? For example, the graph of $y=x^2$ looks like this.

As you can see, it is a curved graph. To write the graph, plot the coordinates of $x$ and $y$ on the graph. Then we can graph a quadratic function by connecting the coordinates with a smooth curve.

The graph of $y=ax^2$ will always pass through the origin. The quadratic function of the graph of $y=ax^2$ passes through the coordinates of $(0,0)$.

Note that the curve of the graph of a quadratic function is the same as the trajectory of a ball thrown. In fact, when throwing an object into the air, the speed can be calculated with a quadratic function.

Earlier, we explained that the trajectory of a rocket is calculated by a quadratic function. The reason for this is that rockets fly in the air. Also, in the past, quadratic functions were used to calculate the trajectory of cannon fire. Quadratic functions are used in everything, including science technology and warfare.

The line in the graph of a quadratic equation is called a parabola. Any object in the air will follow a parabola and will have the same curve as the quadratic function.

The Graph Shape Changes with the Value of $a$

When writing a graph of $y=ax^2$, consider that the shape of the graph depends on the value of $a$. The larger the absolute value of $a$, the steeper the graph. On the other hand, the smaller the absolute value of $a$, the looser the graph becomes.

The shape of the graph depends on the value of $a$, as shown below.

All graphs pass through the origin $(0,0)$. It is also common to have a parabolic curve. The difference is that the shape of the graph changes depending on the value of $a$.

Convex Down at $a>0$ and Convex Up at $a<0$

Also, there is another important point. It is that the orientation of the graph changes depending on whether the value of $a$ is a positive or negative number.

In the graph of a quadratic function, if $a$ is a positive number ($a>0$), it will be convex downward. On the other hand, if $a$ is a negative number ($a<0$), it will be convex upward. Specifically, it looks like the following.

In a quadratic function, the value of $x$ is squared. So whether the value of $x$ is positive or negative, the answer will always be positive.

But in a quadratic function, you multiply the value of $a$. Therefore, if the value of $a$ is positive, the larger the value of $x$, the larger the value of $y$. On the other hand, if the value of $a$ is negative, the larger the value of $x$ is, the smaller the value of $y$.

The Rate of Change in Quadratic Functions Is Different

In a quadratic function, we may need to find the rate of change. When calculating a linear function, the rate of change is very important. This is because of the following.

  • The rate of change = the slope

By calculating the rate of change, we can get the slope $a$.

However, linear and quadratic functions have different ideas about the rate of change. For a linear function, the rate of change is always the same. In quadratic functions, on the other hand, the rate of change varies from place to place.

When calculating the rate of change, there is the following formula.

We don’t need to remember this formula for linear functions. As mentioned earlier, the rate of change and the slope have the same meaning. On the other hand, what about quadratic functions? Even with quadratic functions, there is no point in remembering the rate of change formula.

The rate of change is, in essence, how much the value of $y$ increases when the value of $x$ increases by 1. If you understand this definition, you don’t need to remember the rate of change formula. We can use the properties of the ratio. It looks like the following.

By using the properties of ratios, you can come up with a rate of change. For example, we have the following.

  • When the value of $x$ is increased by 1, the value of $y$ is increased by 2: the rate of change is 2
  • When the value of $x$ is increased by 1, the value of $y$ is increased by -4: the rate of change is -4
  • When the value of $x$ is increased by 7, the value of $y$ is increased by 3: the rate of change is $\displaystyle\frac{3}{7}$

For example, if the value of $x$ is increased by 7 and the value of $y$ is increased by 3, the ratio is $7:3=1:a$. Solving for this, we have $a=\displaystyle\frac{3}{7}$. If you remember a formula that is used infrequently and is not important, you are sure to forget it. Rather, you have to understand the principles of why it happens and come up with a value without a formula.

In mathematics, there are situations where we have to remember formulas and situations where we don’t. For the rate of change, there is no point in remembering the formula.

-The Rate of Change in Quadratic Equations

As mentioned earlier, the rate of change in quadratic functions varies with the coordinates. How can we consider the rate of change in a quadratic function? As an example, let’s calculate the rate of change in the graph of $y=x^2$ for the following.

  • When the value of $x$ changes from 1 to 2
  • When the value of $x$ changes from -3 to -1

When the value of $x$ changes from 1 to 2, the value of $y$ increases by 3. When $x$ increases by 1, $y$ increases by 3, so the rate of change is 3.

On the other hand, when the value of $x$ changes from -3 to -1, the value of $y$ increases by -8. When $x$ increases by 2, $y$ increases by -8. The ratio of change is -4 because $2:-8=1:a$.

Find a Linear Function Through Two Points

Why do we need to find the rate of change in quadratic functions? That’s because we have to solve a problem that combines quadratic and linear functions, as shown below.

The rate of change in the quadratic function changes, but the rate of change in the linear function does not change. Therefore, for a quadratic function, for a line connecting the coordinates of two points, the rate of change is always the same. This is because the slope is the rate of change.

Finding the rate of change in a quadratic function is the same meaning as finding the slope of a line connecting two coordinates. For reference, in the previous graph, $(-1,1)$ and $(3,9)$ are the intersection points. As a result of $x$ is increased by 4, $y$ is increased by 8.

Applying “Increase in $x$: increase in $y$ = 1 : $a$,” then $4:8=1:a$. Solving for this, we get $a=2$.

And if we check the line through the coordinates of the two points, we can see that it passes through the coordinates of $(0,3)$. That is, the intercept is 3. Therefore, the linear function of the line passing through the intersection of the two points is the following.

  • $y=2x+3$

Thus, we can find a linear function for a line connecting two coordinates of a quadratic function. We should be able to solve these math problems as they are frequently given as developmental problems for quadratic functions.

Exercises: Quadratic Function Graph Problems

Q1: Solve the following problems.

We have the graph of $y=ax^2$ (1) and the graph of $y=ax+b$ (2). The origin is O, and the intersection of (1) and (2) is A and B. The $x$ coordinate of A is -2 and the $x$ coordinate of B is 6. Also, set C as the intersection of the line in (2) and the $x$ coordinate.

  1. For the graph in (1), when the value of $x$ increased from -6 to -2, the value of $y$ increased by -16. Let’s find the value of $a$.
  2. Find the equation for the line in (2).
  3. Find the area of △AOB.
  4. Take the point D on the graph in (1). If the area of △COD is 27, find the $x$ coordinates of the point D.

A1: Answers.

In quadratic problems, it is not always just questions about quadratic functions. It is usually a complex of many problems, such as linear functions, simultaneous equations, graphs, etc., related to quadratic functions, as in this problem.

(a)

How can we get an expression for a quadratic function from the rate of change? To do this, consider the amount of increase in $x$ and $y$.

When the value of $x$ is increased from -6 to -2, the value of $x$ is increased by 4. How much does the value of $y$ increase in this case? Substitute into the formula $y=ax^2$.

If the value of $x$ is -6, then $y=a×(-6)^2=36a$. If the value of $x$ is -2, then $y=a×(-2)^2=4a$. Since the value of $x$ has gone from 36a to 4a, $y$ has been increased by -32a.

Since we already know that the change in $y$ is -16, we can calculate the following

$-32a=-16$

$a=\displaystyle\frac{-16}{-32}$

$a=\displaystyle\frac{1}{2}$

Thus, the function in (1) is found to be $y=\displaystyle\frac{1}{2}x^2$.

(b)

Since $y=\displaystyle\frac{1}{2}x^2$, we know the coordinates of A and B. By substituting the value of $x$, we can see that we get the following coordinates, respectively.

  • A$(-2,2)$
  • B$(6,18)$

If we know the two coordinates, we can solve the linear equation by a simultaneous equation. That is, we know the equation in (2). By substituting $y=ax+b$, we solve the following simultaneous equation.

$\begin{eqnarray} \left\{\begin{array}{l}2=–2a+b\\18=6a+b\end{array}\right.\end{eqnarray}$

By solving this system of linear equations, we can get the equation for a linear function.

However, it is also possible to get the equation (2) by another method. That is, using the rate of change. Since we have learned about the rate of change in quadratic functions, we will use the rate of change to come up with an equation (2) in this problem.

Comparing point A and point B, what is the rate of change: when moving from A $(-2,2)$ to B $(6,18)$, the value of $x$ is increased by 8 and the value of $y$ is increased by 16. The ratio of change represents how much the value of $y$ increases when the value of $x$ increases by 1. Therefore, we have the following ratio.

  • $8:16=1:a$

Using the properties of the ratio, we can calculate the following.

$16=8a$

$a=2$

Thus, we find that the slope (ratio of change) of the linear function in (2) is 2. Therefore, the formula for the linear function is $y=2x+b$.

Then let’s get the intercept by substituting A $(-2,2)$ or B $(6,18)$. For example, substituting $(6,18)$ yields the following

$18=2×6+b$

$18=12+b$

$b=6$

Thus, we find that the linear function in (2) is $y=2x+6$.

(c)

How do we calculate the area of △AOB? Let P be the intercept (intersection with the $y$ coordinate) in (2), and we will find the following.

  • $△AOB=△AOP+△BOP$

So, let’s find the areas of △AOP and △BOP. Since equation (2) is $y=2x+6$, the coordinate of P is $(0,6)$. That is, the vertical length of the triangle (line segment PO) is 6.

On the other hand, the $x$ coordinate of A is -2, so the horizontal length of △AOP is 2. The $x$ coordinate of B is 6, so the horizontal length of △BOP is 6. Therefore, we can get the area of each as follows.

  • The area of △AOP: $6×2×\displaystyle\frac{1}{2}=6$
  • The area of △BOP: $6×6×\displaystyle\frac{1}{2}=18$

Adding the area of each triangle gives the answer as follows.

  • $△AOB=6+18=24$

(d)

If there is a point D on (1), then △COD is as follows.

The linear function in (2) is $y=2x+6$. Therefore, the intersection with the $x$ axis can be calculated by substituting $y=0$ as follows.

$0=2x+6$

$-2x=6$

$x=-3$

The coordinate of the point C is $(-3,0)$. That is, the length of CO is 3. On the other hand, what is the vertical length of △COD? The quadratic function of (1) is $y=\displaystyle\frac{1}{2}x^2$. So the coordinate of point D is $(x,\displaystyle\frac{1}{2}x^2)$. That is, the length of the vertical is $\displaystyle\frac{1}{2}x^2$.

The area of the △COD is 27. Applying the formula for the area of the triangle, we can calculate as follows.

$3×\displaystyle\frac{1}{2}x^2×\displaystyle\frac{1}{2}=27$

$x^2=27×\displaystyle\frac{4}{3}$

$x^2=36$

$x=-6,x=6$

Thus for $x=-6$ or $x=6$, the area of △COD is 27.

Read Quadratic Graphs and Solve Equations

The quadratic functions we learn in mathematics are similar in nature to linear functions. However, there is a fair amount to learn and you have to understand the concept of a parabola.

Also, solving quadratic function problems does not necessarily involve only quadratic functions. It is often a complex problem that includes linear equations, simultaneous equations, linear functions, and graphs. So try to use the properties of quadratic functions to solve the problems.

Of particular importance in quadratic functions are the value of $a$ and the rate of change. The value of $a$ will change the shape of the graph of the quadratic function. By calculating the rate of change, we can also calculate the slope of the line through the two coordinates.

Many natural phenomena in the world are quadratic functions. Learn the properties of quadratic functions and try to understand how to draw graphs and solve problems.