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# ZingPath: Linear Relationships and Real World Modeling

## The y-Intercept of a Line with Tables, Graphs, and Equations               Searching for

## Linear Relationships and Real World Modeling

Learn in a way your textbook can't show you.
Explore the full path to learning Linear Relationships and Real World Modeling

### Lesson Focus

#### The y-Intercept of a Line with Tables, Graphs, and Equations

Algebra Foundations

### Learning Made Easy

You will get to learn the exact definition of linear, plus the most important properties of linear relationships–like slope.

### Now You Know

After completing this tutorial, you will be able to complete the following:

• Interpret the meaning of the x-intercept of a linear function using a data table.
• Interpret the meaning of the x-intercept of a linear function using a symbolic representation.
• Interpret the meaning of the x-intercept of a linear function using a graph.

### Everything You'll Have Covered

If a data table models the values of a linear function, how can we use the table to find the x-intercept of the graph of the function? What is the x-intercept of the graph of the linear function whose values are given by the following data table if time is the independent variable and speed is the dependent variable? ~ We can identify the x-intercept of a function from a data table for the function by finding the row where y = 0; the x value in this row gives the coordinates of x-intercept. In the given data table, time is the independent variable (the x variable) and speed is the independent variable (the y variable). We notice that speed is zero at 5 seconds. Therefore, the x intercept of the graph of the linear function with the given data table occurs at (5, 0).

How can we use graphs to determine the x-intercept of the graph of a linear function? What is the x-intercept of the graph given below? ~ The x-intercept of a graph is a point at which the graph crosses the x-axis. Therefore, we need only determine at which point the graph of a linear function crosses the x-axis. In the given graph, we see that the function's graph crosses the x-axis at the point (5, 0); this is the x-intercept of the graph of the function.

How can we use algebra and the equation of a linear function to find the x-intercept of the function? What is the x-intercept of the graph of the linear function whose equation is v(t) = 49 - 9.8t, where t is the independent variable?

~ The x-intercept corresponds to the point where the dependent variable is zero. If a linear function has the equation y(x) = ax +b, we set y = 0 and use inverse operations to solve for x. The solution then gives the x-intercept of the function. In the example, we set v(t) = 0. This gives the equation 0 = 49 - 9.8t, which we then solve for t: 0 = 49 - 9.8t, then 9.8t = 49, and finally t = 49/9.8, t=5. In this case, t is the independent variable, so the x-intercept of the graph of the function is the point (5,0).

In the Animation, John and his friends are doing an experiment in their physics class. They use a machine that throws balls vertically into the air and measure the ball's speed each second as it rises into the air. The data table, graph, and line equation found in the previous problems correspond to a function that models the ball's speed. What is the meaning of the x-intercept of this function in the context of the experiment?

~ The x-intercept corresponds to the time at which the ball's speed is zero. As the ball rises higher into the air, its speed decreases because of gravity. The speed of the ball is zero just before the ball begins to fall back down towards the ground. In this sense, the x-intercept corresponds to the time at which the ball's speed is zero, and this is just before the ball begins to fall back down.

The Animation describes three methods for determining the x-intercept of the graph of a linear function. Which of the three do you find the easiest? Which of the three do you find the most difficult? Using a method of your choice and your answers from the previous section, find the x-intercept of the linear function whose equation is y(x) = -2x +8.

~ Answers will vary. The x-intercept of the graph of the function y(x) = -2x +8 is the point (4, 0). We can see this by noticing that the data table for the function has a 0 for the y-value when x = 4, by noticing that the graph of the function crosses the x-axis at (4, 0), or by noting that x = 4 is a solution of 0 = -2x +8. "

### Tutorial Details

 Approximate Time 2 Minutes Pre-requisite Concepts Students should be able to usea data table, dependent variable, and graph. Course Algebra Foundations Type of Tutorial Animation Key Vocabulary algebraic expression, data table, dependent variable