Problem 10 - Algebra 1

Question

Suppose that bike rentals cost \(\$ 4\) plus \(\$ 1.50\) per hour. Graph the equation. Label the \(y\) -intercept

Step-by-Step Solution

Answer

The line graph passes through y-intercept (0,4), representing a fixed bike rental cost of $4. With each additional hour of rental (x), the cost increases by $1.50, shown as a consistently rising line from the y-intercept.

Step 1: Identify the y-intercept

The y-intercept corresponds to the fixed cost, which is the cost of renting a bike regardless of how long it's used. In the equation given, this cost is $4 so the y-intercept is (0, 4).

Step 2: Identify the slope

The slope of the graph represents the cost per hour of renting the bike, which is $1.50. This is the amount the total cost increases for each additional hour the bike is rented.

Step 3: Plot the y-intercept and draw the line

First, plot the y-intercept on a graph at point (0,4). Since the slope is 1.50, for every 1 unit increase in hours (x-axis) there must be an increase of 1.50 in cost (y-axis). Using this information, draw the line that represents this relationship. Make sure the line passes through the y-intercept.

Step 4: Label the y-intercept

Label the y-intercept. The y-intercept (0,4) should be clearly marked. It corresponds to the cost of renting a bike when it's not used at all, which is $4.

Key Concepts

Understanding the Y-Intercept

The concept of the y-intercept is fundamental in understanding linear equations. The y-intercept is the point where a line crosses the y-axis on a graph. In the context of our exercise, it represents a fixed starting cost, regardless of any other factors. For a bike rental scenario, this is the base fee charged simply for taking the bike, before any hours of use are considered.

In mathematical terms, the y-intercept occurs when the value of the x variable is zero. Thus, it is in the form of (0, b), where "b" is the y-intercept. In the given exercise, this value is $4, which means the y-intercept is (0, 4). This means you will pay $4 just for renting the bike, even if you decide not to ride it at all. The y-intercept gives important information about the initial or fixed condition in any linear situation.

Here’s why you should remember the y-intercept:

• It provides a starting point for graphing a linear equation.
• In many real-world situations, it depicts a fixed fee or charge.

Grasping the Slope of a Line

The slope in a linear equation is a measure of how steep the line is on a graph. It tells you how much the y-value (cost, in this exercise) changes for each unit change in the x-value (hours of bike rental). Slope is calculated by the change in y over the change in x, often termed 'rise over run.'

In our bike rental example, the slope is $1.50, which equates to $1.50 for every hour the bike is rented. The slope is positive, indicating that the longer you rent the bike, the higher the total price will get. The slope forms a crucial part of the linear equation, written as y = mx + b, where m represents the slope.

A steeper slope means:

• A larger rise in cost per unit of time rented.
• A more noticeable incline on the graph.

Graphing a Line with Given Elements

Graphing a line using a linear equation involves two key steps — recognizing the y-intercept and understanding the slope. Once you have these, you can accurately depict the situation on a Cartesian coordinate system.

Start by plotting the y-intercept on the graph. For our equation, the point (0, 4) must be marked on the y-axis. This shows the base charge of the bike rental. Next, use the slope to determine your next point. Since the slope is 1.50, you'll move one unit right on the x-axis (representing one hour of bike rental) and then 1.50 units up on the y-axis (representing the cost increase).

Continue marking points using this method, and then draw a straight line through all of them. This line should extend straight through the y-intercept and illustrate a clear correlation between rental time and cost. Labeling the line and key points such as the y-intercept ensures that the graph clearly communicates this relationship.

Why graphing is beneficial:

• Visualizes relationships between variables.
• Makes it easier to predict values within the context.