Problem 4 - Precalculus : Building Concepts and Connections

Question

Complete them to review topics relevant to the remaining exercises. Find the \(x\) -intercept of \(f(x)=3 x+9\).

Step-by-Step Solution

Answer

The x-intercept of the function is \(x = -3\).

Step 1: Set f(x) to zero

The x-intercept of a function is found where the function equals zero. So we begin by setting \(f(x) = 0\). This gives us the equation \(0 = 3x + 9\).

Step 2: Solve the equation for x

Next, we subtract 9 from both sides of the equation to isolate \(3x\). This results in \(-9 = 3x\). Then, we divide both sides of the equation by 3 to find the value of \(x\). Hence, \(x = -9 / 3 = -3\).

Key Concepts

Linear Functions

Understanding linear functions is fundamental in algebra. These functions represent relationships where changes in one variable lead to proportional and constant changes in another. They can be identified by their straight-line graphs and are typically expressed in the form of the equation, \(f(x) = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In the exercise, \(f(x) = 3x + 9\), 3 is the slope and 9 is the y-intercept. This means that for every unit increase in \(x\), \(f(x)\) or \(y\) increases by 3 units. The y-intercept signifies that when \(x=0\), \(y\) is 9. Understanding this allows us to visualize the graph of the function and predict its behavior across different values of \(x\).

Solving Equations

Solving equations is a staple in algebra, essential for finding unknown values that make an equation true. The process involves manipulating the equation using arithmetic operations to isolate the variable of interest. In the case of finding the x-intercept, we set the output of the function, \(f(x)\), to zero since the x-intercept is the point where the graph of the function crosses the x-axis, thus the y-value is zero. By subtracting \(9\) from both sides in \(0 = 3x + 9\), we eliminate the constant term on the right, simplifying to \(3x = -9\). Afterward, we divide by \(3\) to isolate \(x\), arriving at \(x = -3\), which is our solution. This step-by-step approach ensures a clear pathway to finding the x-intercept, a skill applicable in various areas of mathematics.

Graphing Linear Equations

Graphing a linear equation involves plotting a straight line on the Cartesian plane that represents the equation's solutions. First, identify two points using the slope and y-intercept: the y-intercept, \(b\), which is where the line crosses the y-axis, and another point using the slope, \(m\), which tells us the steepness of the line. For each unit increase along the x-axis, the line will rise by the slope's value along the y-axis. To graph \(f(x) = 3x + 9\), start at the y-intercept (0,9) and use the slope to find a second point – since the slope is 3, go up 3 units and right 1 unit to plot another point. Connect these points to form the line, and where this line crosses the x-axis will visually confirm the x-intercept found algebraically: \(x = -3\). This graphical representation not only solves the provided equation but also aids in comprehending the relationship between the algebraic formula and its geometric interpretation.