Problem 2 Each contain a partial table of ... [FREE SOLUTION]

91影视

Calculus Multivariable

Deborah Hughes-Hallett, Andrew M. Gleason, William G. McCallum

$Math Studyset 91影视 Explanations$ Math

7 Edition

Chapter 12: Problem 2

Each contain a partial table of values for a linear function. Fill in the blanks. $$\begin{array}{|c|c|c|c|} \hline x \backslash y & -1.0 & 0.0 & 1.0 \\ \hline 2.0 & 4.0 & & \\ \hline 3.0 & & 3.0 & 5.0 \\ \hline \end{array}$$

Short Answer

Expert verified

Equation: $ y = x + 2 $; Missing values: $ y(0.0)=2 $.

Step by step solution

Understand the Linear Function

A linear function can be represented by the formula $ y = mx + c $, where $ m $ is the slope and $ c $ is the y-intercept. We need to determine both $ m $ and $ c $ using the given points.

Find the Slope (m)

To find the slope $ m $, we use the formula $ m = \frac{y_2 - y_1}{x_2 - x_1} $. From the table, we use the points (2.0, 4.0) and (3.0, 5.0). Calculating gives $ m = \frac{5.0 - 4.0}{3.0 - 2.0} = 1 $.

Determine the Y-Intercept (c)

Using the slope $ m = 1 $ and one of the known points, let's use (2.0, 4.0). Substitute into the equation $ y = mx + c $: $ 4.0 = 1 \cdot 2.0 + c $. Solving this, we get $ c = 2 $.

Write the Equation of the Line

Now that we have $ m = 1 $ and $ c = 2 $, the equation of the line is $ y = x + 2 $.

Fill in the Missing Values in the Table

Substitute the missing $ x $ values into the equation $ y = x + 2 $ to find the corresponding $ y $ values: - When $ x = 0.0 $, $ y = 0.0 + 2 = 2.0 $- When $ x = 3.0 $, the given $ y $ is already 5.0, hence confirmed.- When $ x = 2.0 $ for the second missing value, it is already established as 4.0.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Slope

The slope of a line in a linear function indicates the steepness and direction of the line. It's represented by the letter $ m $ in the equation of a line, $ y = mx + c $. To understand the slope:

Positive slope: If the slope is positive, the line rises from left to right. As in the example, our slope is $ m = 1 $, making the line incline upwards.
Negative slope: A negative slope means the line falls from left to right.
Zero slope: A slope of zero results in a horizontal line, indicating no rise or fall.

To calculate the slope, use the formula $ m = \frac{y_2 - y_1}{x_2 - x_1} $. This ratio of change in $ y $ to change in $ x $ tells us how much $ y $ increases or decreases as $ x $ increases. In our table, using points (2.0, 4.0) and (3.0, 5.0), we find $ m = 1 $, showing a steady increase by 1 unit of $ y $ for each unit increase in $ x $.
Learning to determine the slope gives valuable insights into the relationship between variables in a function.

Y-intercept

The y-intercept is where the line crosses the y-axis. In terms of the equation $ y = mx + c $, it's symbolized by $ c $. Understanding the y-intercept is crucial:

Starting point: The y-intercept tells you where the line starts on the y-axis when $ x = 0 $.
Graphical representation: At this point, the line intersects the vertical y-axis, marking the value of $ y $ when $ x $ is zero.

To find the y-intercept, take one calculated point, such as (2.0, 4.0), and substitute it into the equation together with the known slope. By rearranging $ y = mx + c $ to solve for $ c $, we find that $ c = 2 $. Thus, when $ x = 0 $, $ y $ is 2, confirming our complete function intercepts the y-axis at $ y = 2 $. This provides the full picture of the line's position and slope relative to the graph.

Equation of a Line

The equation of a line in a linear function is a formula that represents the relationship between $ x $ and $ y $. It's often expressed as $ y = mx + c $, where $ m $ is the slope and $ c $ is the y-intercept. Knowing this form is important:

Describes entire line: The equation holds the essence of the line, portraying its direction and position on the graph.
Predicts values: It allows us to calculate $ y $ for any given $ x $ value, or vice versa.

In our solution, we determined the slope as $ m = 1 $ and the y-intercept as $ c = 2 $. Therefore, the equation becomes $ y = x + 2 $. This lets us fill in any missing table values by substituting $ x $ values into this equation. For example, when $ x = 0 $, substituting gives $ y = 0 + 2 = 2 $. Thus, the line's rule can easily help us uncover unknowns in the function, making it a powerful tool in algebraic contexts.

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91影视

Each contain a partial table of values for a linear function. Fill in the blanks. $$\begin{array}{|c|c|c|c|} \hline x \backslash y & -1.0 & 0.0 & 1.0 \\ \hline 2.0 & 4.0 & & \\ \hline 3.0 & & 3.0 & 5.0 \\ \hline \end{array}$$

Short Answer

Step by step solution

Understand the Linear Function

Find the Slope (m)

Determine the Y-Intercept (c)

Write the Equation of the Line

Fill in the Missing Values in the Table

Key Concepts

Slope

Y-intercept

Equation of a Line

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