Graphs of Linear Functions

Functions can be represented by graphs. When graphed, linear functions become straight lines, with their direction and position determined by the slope and y-intercept.

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Characteristics of the Graph

The graph of a linear function is a straight line, with its direction and position determined by the slope and y-intercept . Understanding slope and y-intercept will enable you to draw graphs of linear functions freely.

Slope and Y-intercept

The slope , as its name suggests, represents how steep the line is and corresponds to $a$ in $y=ax+b$ . The y-intercept represents the y-coordinate of the point where the line intersects the y-axis and corresponds to $b$ in $y=ax+b$ .

It's quickest to understand these by seeing and manipulating them, so use the graph below to change each value and see how the shape of the graph changes. This will help you understand instantly.

Did you understand? Changing the slope alters the steepness of the graph, with a positive slope resulting in an upward-sloping line and a negative slope resulting in a downward-sloping line . The y-intercept represents the y-coordinate of the intersection point with the y-axis, so changing the y-intercept moves the graph up or down .

Also, when the slope is 2, the graph becomes parallel to the gray line $y=2x$ . In other words, graphs with the same slope are parallel to each other .

Slope and Rate of Change

What Does a Slope of 2 Mean?

The example in the sign showed a 10% (0.1) gradient, but let's consider what a slope of 2 means.

For example, let's say we have a linear function $y=2x+2$ with a slope of 2 and a y-intercept of 2, as shown in the graph below. This linear function intersects the y-axis at point $(0, 2)$ because the y-intercept is 2, and passes through point $(1, 4)$ because $y=4$ when $x=1$ . Since $y=2x+2$ , $y$ increases by 2 times the increase in $x$ . This 2 times comes from the slope .

What happens when we increase $x$ by 10? What about when we increase it by 100? That's right. No matter how much we increase $x$ , $y$ always increases by 2 times $x$ .

In the case of a linear function, $y$ always increases at a constant rate relative to the increase in $x$ , and this rate is called the slope .

So... have you noticed? That's right. The rate of change and the slope refer to the same thing. And because the slope/rate of change is constant, the graph becomes a straight line.

Relationship Between the Domains of Two Variables

There may be restrictions on $x$ or $y$ in a linear function $y=ax+b$ . In fact, in the real world, it's rare for variable values to be allowed without limit, isn't it?

For example, we drew a graph $y=5x+10$ above to represent the relationship between temperature $x$ and the number of cold drinks sold $y$ . $x$ represents temperature, but is a value like $x=70$ (°C) possible? The hottest place on Earth is Death Valley in the United States, with a record high temperature of about 53.3°C in 2024. Or, while below freezing might be possible, if $x=-10$ (°C), then $y=-40$ (drinks), meaning the store would somehow be giving away drinks...

Since $y$ is determined by $x$ , when the domain of $x$ is set, the range of $y$ is also determined. Use the graph below to move the domain of $x$ (minimum and maximum values) and the slope, and see how the range of $y$ changes.

Finding the Equation of a Linear Function

So far, we've started with an equation and then drawn its graph or found its domain and range. Next, let's look at how to find the equation from a graph.

Reading from the Graph

Try to find the equation from the linear function graph below.

Since it's a linear function, we need to find $a$ and $b$ in $y=ax+b$ . Also, remember that from the properties of a straight line, we can draw a line if we know the coordinates of two points.

First, if we can find the y-coordinate of the y-intercept (point A), we can directly find $b$ in $y=ax+b$ , so we read the coordinates of the y-intercept. Next, we look for a point B on the line with integer coordinates and read its coordinates from the graph. Finally, we calculate the slope from the increase in $x$ and $y$ from point A to point B.

With this, we know the slope ( $a$ ) and y-intercept ( $b$ ), allowing us to find the linear function equation $y=ax+b$ .

Finding from Conditions Other than Graphs

Look at the following problems:

$y$ is a linear function of $x$ where:

1. The graph is a straight line passing through the point $(1, 2)$ with a slope of $3$ .
2. The rate of change is $-5$ , and $y=3$ when $x=2$ .
3. $y=2$ when $x=-3$ , and $y$ increases by $5$ when $x$ increases by $3$ .
4. The graph is a straight line passing through the point $(0, 5)$ and parallel to the graph of $y=\frac{2}{3}x$ .
5. The graph is a straight line passing through two points $(0, -2)$ and $(4, 1)$ .
6. $y=2$ when $x=-2$ , and $y=8$ when $x=2$ .

As you can see, there are various patterns of problems where we need to find the equation of a linear function from information other than a graph, but essentially, we just need to find the slope $a$ and y-intercept $b$ in $y=ax+b$ . In determining the equation of a linear function, the unknowns to be solved are $a$ and $b$ .

For problems 1-4, we can (almost) directly find either the slope $a$ or y-intercept $b$ , so there's only one unknown left. Therefore, we can solve for the remaining unknown by substituting the given $x$ and $y$ values to create one equation.

On the other hand, in problems 5 and 6, we can't directly know the slope $a$ and y-intercept $b$ . In this case, since there are two unknowns left, we can solve them using the simultaneous equations we learned in the previous unit. (If we can create as many equations as there are unknowns, we can find the solutions for all unknowns.)