Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to graph the equation and approximate any \(x\) - and \(y\) -intercepts. Verify your results algebraically. $$y=4(x+3)-2$$

Step-by-Step Solution

Answer

The y-intercept is (0,10) and the x-intercept is approximately (-2.5, 0), confirmed by both graphical and algebraic means.

Step 1: Plot the Line Using a Graphing Utility

To plot this function, we need to understand the form \(y=mx+b\), where \(m\) is the slope and \(b\) is the y-intercept. For our equation, \(y=4(x+3)-2\), which can also be written as \(y=4x+12-2\) or \(y=4x+10\), the slope \(m\) is \(4\) and the y-intercept \(b\) is \(10\). This means the line begins at the point (0,10) on the y-axis and rises by 4 units for each unit move to the right on the x-axis. Plug in these values into the graphing utility to generate the line.

Step 2: Approximate the x and y Intercepts

The y-intercept is the point (0,10), where the line crosses the y-axis. From the graph, the x-intercept (where the line crosses the x-axis) can be approximated as (-2.5, 0).

Step 3: Verify Intercepts Algebraically

To verify the y-intercept algebraically, set x to 0 in our equation, which gives \(y=4(0+3)-2 = 10\). This confirms the y-intercept. To verify the x-intercept, set y to 0 in the equation and solve for x, giving \(x = -\frac{10}{4} = -2.5\). This confirms our x-intercept.

Key Concepts

Graphing Utility

Using a graphing utility is an invaluable skill when it comes to visualizing algebraic equations, especially linear ones. A graphing utility, such as a graphing calculator or software, allows you to input an equation and see its graph on a coordinate plane.

When you type in the linear equation, the tool plots it by computing and displaying numerous points that satisfy the equation, then connecting them to form a straight line. This visual representation helps to understand different attributes of the line such as its slope, direction, and intercepts.

A plus of using a graphing utility is that you can quickly approximate the points where the line crosses the axes, known as the x-intercept and y-intercept. Moreover, it serves as an excellent cross-verification tool for any algebraic solutions you arrive at.

X-Intercept

The x-intercept of a graph is the point(s) at which the line crosses the x-axis. At these points, the value of y is zero because they lie on the x-axis. To find the x-intercept algebraically, you set the y variable to zero and solve the given equation for x.

For instance, in our equation from the exercise, setting y to zero, you would solve the equation \(0 = 4(x+3) - 2\) to find the x-intercept. After simplification, the x-intercept is \(-2.5\).

In addition to the algebraic method, you can visually approximate the x-intercept using a graphing utility. This not only simplifies the process but also confirms the algebraic result.

Y-Intercept

The y-intercept is where the line crosses the y-axis, which happens at a point where x is zero. To find this algebraically, simply substitute zero for x in the equation and solve for y.

With the equation from the exercise, plugging in zero for x gives us \( y = 4(0+3) - 2 = 10 \), which indicates that the y-intercept is at the point (0,10). This approach of setting x to zero is always used to find the y-intercept of any line.

Graphing the equation with a graphing utility immediately shows the point on the graph where the line intersects the y-axis, further confirming what we found by calculation.

Slope-Intercept Form

The slope-intercept form of a line, given by \(y = mx + b\), is one of the most straightforward ways to write the equation of a line. In this form, \(m\) represents the slope of the line, and \(b\) is the y-intercept.

From our example, the equation can be rewritten in slope-intercept form as \(y = 4x + 10\), with 4 being the slope and 10 being the y-intercept. This tells us for every unit increase in x, y increases by 4 units, which shows the steepness of the line on the graph. Moreover, the y-intercept being 10 means the line crosses the y-axis at the point (0,10).

The slope-intercept form is especially useful because it directly provides both the slope and y-intercept, allowing for a quick sketch of the graph without needing a graphing utility.