Problem 21 - Beginning and Intermediate Algebra

Question

Graph each function. $$h(a)=-2 a+1$$

Step-by-Step Solution

Answer

To graph the function \(h(a) = -2a + 1\), first find the x-intercept by setting \(h(a) = 0\): \(a = \dfrac{1}{2}\). The x-intercept is \(\left(\dfrac{1}{2}, 0\right)\). Next, find the y-intercept by setting \(a = 0\): \(h(0) = 1\). The y-intercept is \((0, 1)\). Plot these points on the coordinate plane and draw a straight line through them to represent the graph of the function.

Step 1: Identify key characteristic points

For the linear function \(h(a) = -2a + 1\), let's first identify the x-intercept and y-intercept. - The x-intercept is a point where the function intersects the x-axis, which means \(h(a) = 0\). - The y-intercept is a point where the function intersects the y-axis, which means \( a = 0 \).

Step 2: Find the x-intercept

To find the x-intercept, set \(h(a) = 0\) and solve for \(a\): \(0 = -2a + 1\) Now, add \(2a\) to both sides of the equation: \(2a = 1\) Finally, divide both sides by \(-2\): \(a = \dfrac{1}{2}\) So, the x-intercept is \(\dfrac{1}{2}\), and the corresponding point is \((\dfrac{1}{2}, 0)\).

Step 3: Find the y-intercept

To find the y-intercept, set \(a = 0\) and calculate \(h(0)\): \(h(0) = -2(0) + 1 = 1\) So, the y-intercept is \(1\), and the corresponding point is \((0, 1)\).

Step 4: Graph the function

We have found two points on the graph of the function: \((\dfrac{1}{2}, 0)\) and \((0, 1)\). To graph the function, plot these points on the coordinate plane and draw a straight line through these points. This line represents the graph of the function \(h(a) = -2a + 1\).

Key Concepts

Linear Equations

Linear equations are among the simplest types of mathematical functions. They describe a line in the coordinate plane. Every linear equation can be written in the form \(y = mx + b\), where \(m\) represents the slope of the line and \(b\) is the y-intercept.
Linear equations are crucial in various fields such as physics, engineering, and economics because they model relationships with a constant rate of change.

• The slope \(m\) indicates the steepness and direction of the line. A positive slope means an upward slant, while a negative slope indicates a downward slant.
• The y-intercept \(b\) is where the line crosses the y-axis.

Understanding linear equations is essential for solving real-world problems and analyzing data patterns.

X-Intercept

The x-intercept of a linear equation is the point where the graph of the equation crosses the x-axis. At this point, the value of \(y\) is zero. To find the x-intercept of the equation \(h(a) = -2a + 1\), set the equation equal to zero.
We'll solve this by setting \(h(a) = 0\) and solving for \(a\):

• \(0 = -2a + 1\)
• Add \(2a\) to both sides: \(2a = 1\)
• Divide by 2 to isolate \(a\): \(a = \dfrac{1}{2}\)

The x-intercept is the point \(\left(\dfrac{1}{2}, 0\right)\). Knowing this point helps when graphing as it provides a precise spot where the line crosses the x-axis.

Y-Intercept

The y-intercept is where the graph of an equation crosses the y-axis. At this point, the value of \(x\) (or \(a\) in this equation) is zero.
For the function \(h(a) = -2a + 1\), substitute \(a = 0\) to find the y-intercept:

• \(h(0) = -2(0) + 1 = 1\)

The y-intercept is \(1\), making the point \((0, 1)\). This is another key point, providing a starting reference to sketch the line on the coordinate plane. The y-intercept enables us to determine the line's vertical position on the graph.

Coordinate Plane

A coordinate plane is a two-dimensional space formed by two perpendicular number lines: the x-axis and the y-axis. It is used to plot points, lines, and curves defined by equations. Each point on this plane is identified by an ordered pair, \((x, y)\), that denotes its position relative to the two axes.
When graphing linear functions like \(h(a) = -2a + 1\):

• The x-axis runs horizontally and is used for the independent variable.
• The y-axis runs vertically and corresponds to the dependent variable.

Understanding how to navigate the coordinate plane is critical for accurately plotting and interpreting data. This concept is foundational for advanced mathematics, including calculus and statistics.

Plotting Points

Plotting points is the process of marking specific locations on the coordinate plane according to their coordinates. For a linear function, identifying at least two points suffices to draw the entire line.
For the function \(h(a) = -2a + 1\), plot the key points:

• The x-intercept \(\left(\dfrac{1}{2}, 0\right)\)
• The y-intercept \((0, 1)\)

Once plotted, draw a straight line through these points using a ruler. This line represents the graph of the given function. Understanding plotting is essential for visualizing mathematical relationships and interpreting the line's behavior across different values.