Problem 5 - Precalculus : Building Concepts and Connections

Question

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

Step-by-Step Solution

Answer

The y-intercept of the function \(f(x) = 3x + 9\) is 9.

Step 1: Understand the definition of y-intercept

The y-intercept of a function is the point where the graph of this function intersects with the y-axis. In other words, it's the value of the function when \(x = 0\). So to find the y-intercept, we need to find the value of the function when \(x = 0\).

Step 2: Substitute x=0 into the equation

So let's substitute \(x=0\) into the function: \(f(x) = 3x + 9\). This gives us \(f(0) = 3*0 + 9\).

Step 3: Simplify to find the y-intercept

Simplifying this results in \(f(0)=9\). Thus, the y-intercept of the function \(f(x) = 3x + 9\) is 9.

Key Concepts

Y-Intercept Definition

The term 'y-intercept' is crucial when understanding the basics of graphing functions. The y-intercept is the point on a graph where the line or curve crosses the y-axis. Specifically, this is where the input value, or the 'x' value, is zero. In essence, the y-intercept answers the question: 'What is the output of the function when the input is zero?'.

To find this point on the graph of a linear function, like in the exercise with the function \(f(x) = 3x + 9\), one should look at where the graph intersects the vertical axis that represents y-values. Mathematically, you determine the y-intercept by substituting zero into the function for x and solving for y. In the given example, when we substitute zero for x, we get \(f(0) = 3(0) + 9\), which simplifies to \(f(0) = 9\). Therefore, the y-intercept is the point \((0, 9)\) on the graph.

Linear Function

Linear functions form the backbone of algebra and provide a great tool for understanding relationships between variables. A linear function is one that has the form \(y = mx + b\), where 'm' stands for the slope, or the rate of change, and 'b' represents the y-intercept, where the line crosses the y-axis.

Graphically, the slope 'm' determines how steep or flat the line is, and 'y-intercept' 'b' determines the exact point on the y-axis where our line will pass through. Every linear function, when graphed, will produce a straight line. For the function \(f(x) = 3x + 9\), the slope is 3 and the y-intercept is 9, which means for every unit increase in 'x', 'y' will increase by 3 units, and the line will cross the y-axis at 9.

Substituting Values in Functions

Substituting values in functions is a method used to evaluate functions for particular values of the variable. This is done by replacing the variable 'x' with a specific value. When dealing with a simple linear function, such as \(f(x) = 3x + 9\), substituting a value is straightforward. For example, if we need to find the value of the function when x is 2, we substitute 2 for every 'x' in the equation, resulting in \(f(2) = 3(2) + 9\), which simplifies to \(f(2) = 6 + 9 = 15\).

Similarly, to find the y-intercept, we substitute 0 for x, because the y-intercept is the point where the function intersects the y-axis and at this point, the value of x must be zero. This method is fundamental in algebra and calculus and can be applied to a wide range of functions beyond the linear ones. Understanding how to properly substitute values allows students to graph functions, predict outcomes, and understand the relationship between variables in an equation.