Explainer: Linear Functions in Different Forms

In this explainer, we will learn how to write linear functions in different forms, such as standard form and slope-intercept form.

We know that a linear relationship between two quantities is characterized by a constant rate of change between both quantities. This type of relationship is a function since every input gives exactly one output. A function that has a constant rate of change is called a linear function. As the name suggests, its graph is a straight line; it is a straight line because the rate of change is constant. Inversely, if the graph of a function is a straight line, then it has a constant rate of change.

The rate of change for a function 𝑓(π‘₯) is defined as the rate of the change in the value of the function to the corresponding change in π‘₯: rateofchange=changein𝑓(π‘₯)changeinπ‘₯.

If 𝑓 is graphed on a coordinate plane, then the 𝑦-coordinate of any point on the graph gives the value of 𝑓 when the input is the π‘₯-coordinate of this point. The rate of change is then simply the rate of the change in 𝑦 to the change in π‘₯ between the two points.

Let us take an example. Consider a linear function 𝑓. We know that 𝑓(1)=2and𝑓(5)=4.

The rate of change between these two values of π‘₯ is then given by rateofchange=changein𝑓(π‘₯)changeinπ‘₯=𝑓(5)βˆ’π‘“(1)5βˆ’1=4βˆ’25βˆ’1=24=12.

Let us look now at the graph of this linear function with 𝑦=𝑓(π‘₯).

We see that any variation in 𝑦 with respect to the variation in π‘₯ can be decomposed in β€œunit variations” represented here by the blue triangles. This is the representation of the rate of change: a change of 1 in π‘₯ leads to a change of 12 in 𝑦, so a change of 4 in π‘₯ leads to a change 4 times bigger than 12 in 𝑦, that is, 2.

We see that, for any two points lying on a straight line, the difference in their 𝑦-coordinates is proportional to the difference in their π‘₯-coordinates. The rate of change is actually the coefficient of proportionality between the change in 𝑦 and the change in π‘₯; it is the slope of the line that represents the function. Note that it can be either positive or negative (or zero, when the line is horizontal, representing a constant function).

Let us now see how to use this result to find an equation between the 𝑦-coordinates and the π‘₯-coordinates of all the points lying on the line in our example above. What do we know about any other point 𝑀(π‘₯𝑀,𝑦𝑀) lying on the same line?

We know that the change in the 𝑦-coordinates between 𝑀 and 𝐴 is given by multiplying the change in the π‘₯-coordinates by the rate of change, that is, by 12. Therefore, we can write π‘¦π‘€βˆ’π‘¦π΄=12(π‘₯π‘€βˆ’π‘₯𝐴).

Since π‘₯𝐴=1 and 𝑦𝐴=2, we have π‘¦π‘€βˆ’2=12(π‘₯π‘€βˆ’1).

We can expand the brackets and find that π‘¦π‘€βˆ’2=12π‘₯π‘€βˆ’12.

Adding 2 to each side then gives us 𝑦𝑀=12π‘₯π‘€βˆ’12+2, which is 𝑦𝑀=12π‘₯𝑀+32.

Since 𝑀 can be any point lying on the line, we can write the equation in a more general form, dropping the subscript 𝑀: 𝑦=12π‘₯+32.

This is the equation of a line: it gives the relationship between the 𝑦- and π‘₯-coordinates of any point lying on the line. It is also the function rule for 𝑓 with 𝑓(π‘₯)=𝑦.

The general form of the equation of a straight line is 𝑦=π‘šπ‘₯+𝑏, where π‘š is the slope of the line and 𝑏 is a constant.

Let us look at the meaning of the constant 𝑏. We find that when π‘₯=0, the equation gives 𝑦=π‘šβ‹…0+𝑏, which simplifies to 𝑦=𝑏.

Therefore, 𝑏 is the 𝑦-coordinate of the point where the line intersects the 𝑦-axis. For this reason, 𝑏 is called the 𝑦-intercept of the line. We can check this on the graph of our line 𝑦=12π‘₯+32; the line intersects the 𝑦-axis at the point ο€Ό0,32.

Let us summarize our findings.

Equation of a Line

The equation of a line is generally written in the form 𝑦=π‘šπ‘₯+𝑏, where π‘š is the slope of the line and 𝑏 is the 𝑦-intercept.

If the line passes through two points of coordinates (π‘₯1,𝑦1) and (π‘₯2,𝑦2), then π‘š is π‘š=𝑦2βˆ’π‘¦1π‘₯2βˆ’π‘₯1.

This can be expressed as π‘š=verticalchangehorizontalchange.

The slope π‘š gives the change in 𝑦 when π‘₯ increases by +1.

The line intersects the 𝑦-axis at the point of coordinates (0,𝑏).

Let us now look at some questions to check our understanding.

Example 1: Finding the Slope of a Line from Its Equation

What is the slope of the line of the equation 𝑦=8π‘₯βˆ’3?

Answer

Recall that the slope–intercept form of the equation of a straight line is 𝑦=π‘šπ‘₯+𝑏, where π‘š is the slope of the line and 𝑏 is the 𝑦-intercept. Here, the equation of the line is 𝑦=8π‘₯βˆ’3.

Comparing both, we find that π‘š=8.

Hence, the slope of this line is 8.

Example 2: Finding the 𝑦-Intercept of a Line from Its Equation

What is the 𝑦-intercept of the line of the equation 𝑦=5π‘₯+7?

Answer

Recall that the slope–intercept form of the equation of a straight line is 𝑦=π‘šπ‘₯+𝑏, where π‘š is the slope of the line and 𝑏 is the 𝑦-intercept. Here, the equation of the line is 𝑦=5π‘₯+7. Comparing both, we find that 𝑏=7.

Hence, the 𝑦-intercept is 7.

Example 3: Finding the Slope of a Line from a Graph

What is the slope of the line shown in the figure?

Answer

Recall that the slope is given by the ratio of the vertical change to the horizontal change between the two points. To find the slope of a line graphically, we need to work out the changes between two well-defined points.

We see that, between the two points of coordinates (0,10) and (5,0), a horizontal change of +5 in π‘₯ gives rise to a change of βˆ’10 in 𝑦. Hence, we have π‘š=βˆ’105=βˆ’2.

The slope of the line shown in the figure is βˆ’2.

We can check that our result is correct by checking that when π‘₯ increases by 1, 𝑦 decreases by 2.

Example 4: Finding the 𝑦-Intercept of a Line from a Graph

What is the 𝑦-intercept of the line shown in the figure?

Answer

The 𝑦-intercept of a line is the 𝑦-coordinate of the point where the line intersects the 𝑦-axis.

Here, the line intersects the 𝑦-axis at the point of coordinates (0,10). Hence, the 𝑦-intercept is 10.

Example 5: Matching the Equation of a Linear Function to Its Graph

Which of the following equations represents the function drawn on the graph?

  1. 𝑓(π‘₯)=βˆ’2π‘₯βˆ’6
  2. 𝑓(π‘₯)=βˆ’2π‘₯+6
  3. 𝑓(π‘₯)=2π‘₯+6
  4. 𝑓(π‘₯)=2π‘₯βˆ’6
  5. 𝑓(π‘₯)=2π‘₯2βˆ’6

Answer

The graph of the function shown is a straight line. Therefore, it is a linear function and its equation is of the form 𝑓(π‘₯)=π‘šπ‘₯+𝑏, where π‘š is the slope and 𝑏 is the 𝑦-intercept of the line that represents 𝑓.

Hence, we can already eliminate option E, which is not a linear function.

In the equations in options A and B, π‘š is βˆ’2, which implies that the lines that represent them have negative slopes. A negative slope means that when π‘₯ increases, 𝑦 decreases. We see that it is not the case here as 𝑦 increases when π‘₯ increases (the line β€œgoes up” from left to right).

Hence, we can eliminate options A and B.

We can quickly check that the slope of the line shown here is indeed 2, as given in the equations in options C and D. A slope of 2 means that 𝑦 increases by 2 when π‘₯ increases by 1.

We find that this is the case indeed.

The difference between the equations in options C and D is the value of 𝑏. Remember that a line of the equation 𝑦=π‘šπ‘₯+𝑏 intersects the 𝑦-axis at the point of coordinates (0,𝑏). Here, the line intersects the 𝑦-axis at the point of coordinates (0,βˆ’6). Hence, the right equation of the function represented by this line is 𝑓(π‘₯)=2π‘₯βˆ’6.

That is, option D.

Key Points

  1. The slope–intercept form of the equation of a line is of the form 𝑦=π‘šπ‘₯+𝑏, where π‘š is the slope of the line and 𝑏 is the 𝑦-intercept.
  2. A line is the graph of a linear function of the equation 𝑓(π‘₯)=π‘šπ‘₯+𝑏.
  3. The slope of a line, π‘š, is equal to the rate of change of the function. The rate of change is given by 𝑦2βˆ’π‘¦1π‘₯2βˆ’π‘₯1 between two points of coordinates (π‘₯1,𝑦1) and (π‘₯2,𝑦2). Graphically, it is π‘š=verticalchangehorizontalchange.
  4. The slope π‘š gives the change in 𝑦 when π‘₯ increases by +1.
  5. The line intersects the 𝑦-axis at the point of coordinates (0,𝑏).

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