Question

a. Graph the line corresponding to the equation \(y=-0.5 x+3\) by graphing the points corresponding to \(x=0,1,\) and 2 . Give the \(y\) -intercept and slope for the line. b. Check your graph using the How a Line Works applet. c. How is this line related to the line \(y=0.5 x+3\) of Exercise \(12.76 ?\)

Short answer

2. What is the y-intercept and slope of the line \(y=-0.5x+3\)? 3. How is the line \(y=-0.5x+3\) related to the line \(y=0.5x+3\)?

Step-by-step solution

Calculate the y-values for x=0, 1, and 2

To graph the line, we need to first calculate the y-values corresponding to x=0, 1, and 2 in the equation \(y=-0.5x+3\). For \(x=0\): \(y=(-0.5)(0)+3=0+3=3\) For \(x=1\): \(y=(-0.5)(1)+3=-0.5+3=2.5\) For \(x=2\): \(y=(-0.5)(2)+3=-1+3=2\)

Graph the line

Plot the points \((0,3)\), \((1,2.5)\), and \((2,2)\) on the coordinate plane to graph the line corresponding to the equation \(y=-0.5x+3\).

Identify the y-intercept and slope

The y-intercept can be identified from the graph as the point where the line intersects the y-axis. In our case, it is \((0,3)\). The slope can be found using the formula \(\frac{change \ in \ y}{change \ in \ x}\). Using the points \((0,3)\) and \((1,2.5)\), the slope can be calculated as: Slope = \(\frac{2.5-3}{1-0}=\frac{-0.5}{1}=-0.5\) So, the y-intercept is 3 and the slope is -0.5.

Analyze the relation between the given line and line of Exercise 12.76

In this part, we need to analyze the relation between the line \(y=-0.5x+3\) (line 1) and the line \(y=0.5x+3\) (line 2) from Exercise 12.76. We can see that the y-intercept of both lines is the same (3), which means that both lines intersect the y-axis at the same point. However, their slopes have opposite signs which results in opposite directions for the rising or falling of the lines. Line 1 has a negative slope, which means it is falling as x increases. On the other hand, Line 2 has a positive slope, which means it is rising as x increases. Therefore, we can conclude that the two lines are reflections of each other over the x-axis, which means they have the same y-intercept but opposite slopes.

Key concepts

Y-Intercept

Understanding where a line crosses the y-axis is a fundamental concept in graphing linear equations. The y-intercept is the point at which the line intersects the y-axis of a coordinate plane. In other words, it's where the value of x is zero. For the linear equation given, which is \(y = -0.5x + 3\), the y-intercept is found by setting x to zero, yielding the point \((0,3)\). This means no matter what our line looks like, it must pass through the point \((0,3)\) on the coordinate plane.

The y-intercept is not just a point on a graph; it can often provide insights into real-world situations if the context of the data is known. For instance, in economic terms, the y-intercept might represent a starting value or fixed cost before other variables come into play.

Slope

The slope of a line indicates its steepness and direction on a coordinate plane. It is calculated as the change in y over the change in x, often remembered by the formula \(m = \frac{\Delta y}{\Delta x}\) where \(m\) represents the slope. The slope tells us how much y increases or decreases for a one-unit increase in x. For our equation \(y = -0.5x + 3\), the slope is -0.5. This negative value means that for every step to the right on the x-axis, the line goes down by 0.5 units on the y-axis.

The concept of slope is pivotal in understanding the behavior of a linear equation. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls. A slope of zero indicates a horizontal line, showing no change in y as x changes, and an undefined slope (resulting from a division by zero) corresponds to a vertical line, where x remains constant.

Coordinate Plane

The coordinate plane is a two-dimensional surface where each point is identified by its x (horizontal) and y (vertical) coordinates. It consists of two perpendicularly intersected lines called axes, creating four quadrants.

Plotting Points

When graphing linear equations, we plot points defined by their (x,y) coordinates and then connect them to get a line. For instance, in our exercise to plot the line for \(y = -0.5x + 3\), points like \((0,3)\), \((1,2.5)\), and \((2,2)\) are graphed on the coordinate plane.

Always label and number your axes correctly, as this helps in accurately representing the relationships between variables in the form of geometric figures like lines, curves, and shapes. This visual representation can be a powerful tool for understanding and communicating mathematical concepts.

Line Equation

A line equation is a mathematical way to describe a straight line on a coordinate plane. It correlates any x-value with its corresponding y-value on the line. The most common form of a line equation is the slope-intercept form, which is written as \(y = mx + b\). Here, \(m\) stands for the slope, and \(b\) represents the y-intercept of the line.

By analyzing this form, one can quickly identify both the slope and y-intercept: in our equation, \(y = -0.5x + 3\), \(-0.5\) is the slope, and \(3\) is the y-intercept. The line equation is a clear, concise way to convey all the information needed to graph a line, including direction, steepness, and starting position. It's a fundamental tool for algebra students to master, eventually leading to understanding more complex mathematical functions.