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Chapter 3: Problem 11

Graph the given functions. $$y=\frac{1}{2} x-3$$

Short Answer

Expert verified
Plot the points (0, -3) and (1, -2.5), then draw a line through them.

Step by step solution

01

Understand the Function Form

The given function is a linear equation in the standard slope-intercept form, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this case, \(m = \frac{1}{2}\) and \(b = -3\).
02

Identify the Y-Intercept

The y-intercept of a linear function is the point where the graph intersects the y-axis. For the equation \(y = \frac{1}{2} x - 3\), the y-intercept is -3. This means the graph will pass through the point (0, -3).
03

Use the Slope to Find Another Point

The slope \(\frac{1}{2}\) indicates that for every 1 unit increase in \(x\), \(y\) increases by \(\frac{1}{2}\). Starting from the y-intercept (0, -3), move right 1 unit to \(x = 1\), then move up \(\frac{1}{2}\) unit to find the next point at (1, -2.5).
04

Plot Points on the Coordinate Plane

Plot the y-intercept (0, -3) and the point (1, -2.5) on a coordinate plane. These points are used to draw the line of the function.
05

Draw the Line

Draw a straight line through the points (0, -3) and (1, -2.5). This line represents the graph of the function \(y = \frac{1}{2} x - 3\). Extend the line across the plane, as the line continues indefinitely in both directions.

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Key Concepts

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

Slope-Intercept Form
Understanding how linear equations can be expressed is important when graphing them. The slope-intercept form of a linear equation is written as:
  • \(y = mx + b\)
This equation provides us with two crucial components needed for graphing:
  • The "slope" \(m\), which shows how steep the line is.
  • The "y-intercept" \(b\), which tells us where the line crosses the y-axis.
In the equation \(y = \frac{1}{2}x - 3\), \(m = \frac{1}{2}\) and \(b = -3\). The slope means for every increase of 1 in \(x\), \(y\) will increase by \(\frac{1}{2}\). The y-intercept tells us the line will cross the y-axis at \((0, -3)\). Knowing these parts helps to swiftly draw the graph.
Finding Y-Intercept
When dealing with linear functions, the y-intercept is a key point, marking where the line crosses the y-axis. In the slope-intercept form \(y = mx + b\), the \(b\) directly represents the y-intercept. For the function \(y = \frac{1}{2}x - 3\), the y-intercept is
  • -3
The line will intersect the y-axis exactly at the point \((0, -3)\). No matter what the slope is, as long as we have the y-intercept, we start plotting from this point. The y-intercept serves as a reference point, from which we can plot additional points using the slope.
Using Slope to Plot Points
The slope of a line indicates its direction and steepness. It is expressed as a ratio or fraction in the slope-intercept equation. In \(y = \frac{1}{2}x - 3\), the slope \(\frac{1}{2}\) tells us:
  • For every 1 unit move in the positive x-direction, move \(\frac{1}{2}\) unit up in the y-direction.
Starting from the y-intercept point \((0, -3)\), move 1 unit to the right to \((1, y)\). Change in y is \(\frac{1}{2}\), so the new point is \((1, -2.5)\). Plot this point on the graph along with the y-intercept. This gives you a reference for the line, ensuring accuracy in slope representation.
Coordinate Plane
The coordinate plane is a two-dimensional surface where we plot and visualize mathematical functions. It's made up of two axes:
  • The x-axis (horizontal)
  • The y-axis (vertical)
These axes divide the plane into four quadrants. Each point on the plane is identified by an ordered pair \((x, y)\). For graphing lines like \(y = \frac{1}{2}x - 3\), mark the points \((0, -3)\) (y-intercept) and \((1, -2.5)\) (second point using slope). With these points, draw a line to extend the graph across the plane. This line visually represents the infinite set of solutions that satisfy the linear equation.

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Most popular questions from this chapter

In Exercises \(37-66,\) graph the indicated functions. Plot the graphs of (a) \(y=x^{2}-x+1\) and \((b) y=\frac{x^{3}+1}{x+1}\)

In Exercises \(37-66,\) graph the indicated functions. Plot the graphs of \(y=x\) and \(y=|x|\) on the same coordinate system. Explain why the graphs differ.

Answer the given questions. If the point \((a, b)\) is in the second quadrant, in which quadrant is \((a,-b) ?\)

Graph the indicated functions. The voltage \(V\) across a capacitor in a certain electric circuit for a \(2-s\) interval is \(V=2 t\) during the first second and \(V=4-2 t\) during the second second. Here, \(t\) is the time (in s). Plot \(V\) as a function of \(t\).

Graph the indicated functions. The power \(P\) (in \(\mathrm{W} / \mathrm{h}\) ) that a certain windmill generates is given by \(P=0.004 v^{3},\) where \(v\) is the wind speed (in \(\mathrm{km} / \mathrm{h}\) ). Plot the graph of \(P\) vs. \(v\).
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