The Rate of Change (a) and the y-Intercept (b)

The equation of a linear function is represented by:

||f(x) = \color{red}{a}x + \color{blue}{b}||

where |\color{red}{a}| and |\color{blue}{b}| are real numbers.

In a relationship between two variables, the rate of change (usually represented by the letter |a|) is the relationship between two corresponding changes of the variables.

||\text{Rate of change} = \dfrac{\text{Change in the dependent variable}}{\text{Change in the independent variable}}||

It can be simplified by the following expression.

||a=\dfrac{\Delta y}{\Delta x} = \dfrac{y_2 - y_1}{x_2 - x_1}||

Where |(x_1,y_1)| and |(x_2,y_2)| are two distinct points on the line and |\Delta| is the Greek letter delta representing a change.

In analytical geometry, the rate of change is called the slope of a line.

The |y|-intercept, also called the initial value and denoted |b,| corresponds to the value of the dependent variable |(y)| when the value of the independent variable |(x)| is |0.| Plotted on a graph, the |y|-intercept is the point of intersection between the line and the |y|-axis.

Examples of equations and their corresponding rate of change.

To determine the rate of change and the |y|-intercept, |y| must be isolated on one side of the equation. In other words, the equation must be in the form |y=ax+b|. If it is not the case, we need to isolate the variable |y.|

Here are three examples where we are looking for the rate of change and the |y|-intercept.

Example 1: ||\begin{align} 2y &= 4x+8 \\ \frac{2y}{\color{green}{2}} &= \frac{4x}{\color{green}{2}} + \frac{8}{\color{green}{2}} \\ y &= \color{red}{2}x+\color{blue}{4} \end{align}||

The rate of change of the equation is |\color{red}{2}| and the |y|-intercept is |\color{blue}{4}.|

Example 2: ||\begin{align} y+6x &= 7 \\ y+6x \color{green}{-6x} &= 7 \color{green}{-6x} \\ y &= 7-6x \\ y &= \color{red}{-6}x+\color{blue}{7} \end{align}||

The rate of change of the equation is |\color{red}{-6}| and the |y|-intercept is |\color{blue}{7}.|

 

Example 3: ||\begin{align} \frac{y}{2}-\frac{6x}{5} &= -3 \\ \frac{y}{2}-\frac{6x}{5} \color{green}{+\frac{6x}{5}} &= -3 \color{green}{+\frac{6x}{5}}\\ \frac{y}{2} &= -3 + \frac{6x}{5} \\ \frac{y}{2} \color{green}{\times 2} &= \left(-3 + \frac{6x}{5}\right) \color{green}{\times 2} \\ y &= -6+\frac{12x}{5} \\y &= \color{red}{\frac{12}{5}}x \color{blue}{-6} \end{align}||

The rate of change of the equation is |\color{red}{\dfrac{12}{5}}| and the |y|-intercept is |\color{blue}{-6}.|

To calculate the rate of change and the initial value, we need to follow these 4 steps.

1. Identify the dependent and independent variables.

2. Choose 2 points on the line or 2 ordered pairs from the table of values.

3. Apply the formula for the rate of change. ||a=\dfrac{\Delta y}{\Delta x}=\dfrac{y_2 - y_1}{x_2 - x_1}||

4. Replace |a| with the value calculated in the previous step and |x| and |y| with the coordinates of a point, and then isolate |b.|

Note: sometimes, step 4 is not necessary to get the |y|-intercept, since it can be found directly in the graph or the table of values.

In the following example, we consider that the individual’s consumption habits are constant.

• How much money does the individual save per month? (rate of change)

• How much did they have saved at the start of the study? (|y|-intercept)

What is the rate of change |(a),| the initial value |(b),| and the equation of the straight line illustrated in the following Cartesian plane?

Marc has just opened a print shop. He bought a high-performance photocopier. He decided on the costs below for using the photocopier.

• What is the price per photocopy?

• What is the initial price?

Ginette removes the plug from her bathtub, then she observes the water level in the bath every 2 minutes. The following is the data she collects.

• How quickly does the water level in the tub drop? (rate of change)

• What was the water level in the bath when Ginette removed the plug? (|y|-intercept or initial value)

At the start of his last trip, Justin's car had |125\ 000| km on the odometer. He maintained an average speed of |100| km/h during the trip. What will be the total mileage on his car after driving for |22| hours?

If the price of gasoline was |\$ 1.12\text{/L}| when I went to get gas, how many litres did I put in my car's fuel tank if I paid |\$ 56?|

Rudolph went camping. He wakes up at 7:00 a.m. and notices that the temperature inside his tent is 15ºC. At 11:00 a.m., he notices that the temperature inside his tent is 21ºC.

• How many degrees on average did the temperature inside his tent rise per hour during the morning?

• What is the initial value (|y|-intercept) and what does it represent in this situation?

A 100-litre water tank empties at a constant rate in 8 minutes.

Represent the situation on a graph, then give the rate of change, the initial value, and the equation representing the situation.