Problem 35 - Algebra Form and Function

Question

If \(a\) is a constant, does the equation \(y=a x+5 a\) define \(y\) as a linear function of \(x ?\) If so, identify the slope and vertical intercept.

Step-by-Step Solution

Answer

If so, identify the slope and the vertical intercept. Answer: Yes, the equation \(y = ax + 5a\) represents a linear function of x. The slope (m) is \(a\) and the vertical intercept (b) is \(5a\).

Step 1: Identify the type of function

Check the equation \(y = ax + 5a\) and observe if it's in the form of a linear equation, i.e., \(y = mx + b\).

Step 2: Compare the given equation with the standard linear equation

The given equation is \(y = ax + 5a\). Comparing it to the standard linear equation form, \(y = mx + b\), we can see that \(m = a\) and \(b = 5a\). Since \(a\) is a constant, this equation defines a linear function of \(x\).

Step 3: Identify the slope and vertical intercept

Now that we have established that this is a linear function, we can identify the slope (m) and the vertical intercept (b). The slope is given by the coefficient of x, which in this case is \(a\). And the vertical intercept is the constant term, which is \(5a\). Slope: \(m = a\) Vertical intercept: \(b = 5a\)

Key Concepts

Understanding the Slope of a Line

The slope of a line is a fundamental concept in linear functions and graphing. It describes how steep the line is and in which direction it is going. To understand the slope, think of it as the 'rise' over 'run'. This means:

• 'Rise' refers to how much the line goes up or down as you move along it.
• 'Run' represents the horizontal distance the line travels.

The slope is represented by the letter \(m\) in the linear equation \(y = mx + b\). When calculating the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) on a line, use the formula:\[m = \frac{y_2 - y_1}{x_2 - x_1}\]
A positive slope indicates the line is rising, while a negative slope shows it is falling. If the slope is zero, the line is horizontal, and if it is undefined, the line is vertical.
In the given equation \(y = ax + 5a\), the slope \(m\) is \(a\). This confirms the direct relationship between the slope and the coefficients in the equation, as long as \(a\) is a constant meaning the function is linear.

Exploring Linear Equations

Linear equations are expressions that create straight lines when graphed on a coordinate plane. They typically come in the general form \(y = mx + b\) where the term \(mx\) identifies the slope and \(b\) is the y-intercept. To verify linearity, ensure the equation can be rearranged to this form.
Features of linear equations include:

• One-degree variables, indicating the highest power of the variable is one.
• A constant rate of change, characterized by the slope.
• A graph that is a straight line.

In the problem equation \(y = ax + 5a\), it aligns perfectly with the general linear form \(y = mx + b\), showing clearly that it is a linear function. Here, \(a\) serves as the slope, showing a steady influence of \(x\), and the constant term \(5a\) indicates the vertical position of the line when \(x = 0\). Linear equations are predictable and straightforward, making them easier to manipulate in many mathematical scenarios.

Function Analysis of Linear Functions

Analyzing linear functions involves understanding the characteristics and implications of the equation's form and graph. Let's dive deeper:
Linear functions, like \(y = ax + 5a\), are straightforward but informative, representing:

• A constant rate of change due to their slope \(a\).
• An initial value or starting point, indicated by the intercept \(5a\).

To analyze such functions, identify the slope and intercept since they dictate the behavior of the line. The slope tells us how changes in \(x\) affect \(y\). On the other hand, the vertical intercept tells where the line will cross the y-axis.
Graphically, it helps to know:

• Where to start drawing (at the y-intercept).
• How steep to make the line (determined by the slope).
• The direction of the line (upwards or downwards based on the slope’s sign).

This kind of analysis helps predict how changes in the input affect the output, providing insight into the real-world meaning of mathematical relationships.