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Chapter 5: Problem 4

Give the values for \(b\) and \(m\) for the linear functions in Exercises 4-9. $$ f(x)=3 x+12 $$

Short Answer

Expert verified
Answer: The slope (m) of the linear function is 3, and the y-intercept (b) is 12.

Step by step solution

01

Identifying the slope (m) and the y-intercept (b)

In the given function, \(f(x) = 3x + 12\), m (slope) and b (y-intercept) are represented by the coefficients of x (3) and the constant (12) respectively.
02

Writing the values of m and b

In the given linear function, \(f(x) = 3x + 12\), the slope (m) is 3, and the y-intercept (b) is 12.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding the Slope
The slope of a linear function measures how steep the line is. It shows the rate at which the line rises or falls as it moves along the x-axis.
This is often represented by the letter \( m \).
  • A positive slope means the line rises as it moves from left to right.
  • A negative slope indicates the line falls as you move from left to right.
  • If the slope is zero, the line is horizontal, showing no rise at all.
In our function \( f(x) = 3x + 12 \), the slope \( m \) is 3. This tells us the line rises three units for every one unit it moves to the right. Imagine a staircase where each step is three units taller than the last. That's what the slope does, it determines the steepness of those steps.
Knowing the Y-Intercept
The y-intercept is where the line crosses the y-axis. It's a key feature of any linear equation, indicating where the function begins when \( x \) is zero.
The y-intercept is represented as \( b \) in the equation \( y = mx + b \).
  • The value of \( b \) is the point in the graph where the line hits the y-axis.
  • Every linear function will have one unique y-intercept.
In our example, \( f(x) = 3x + 12 \), the y-intercept \( b \) is 12. This means the line crosses the y-axis at the point (0, 12). Think of it as the starting point of the line on the graph before any change in \( x \) occurs.
Linear Equations and Their Form
Linear equations are fundamental objects in algebra that describe straight lines. They are usually written in the slope-intercept form, \( y = mx + b \), where:
  • \( m \) is the slope.
  • \( b \) is the y-intercept.
This form makes it easy to quickly find important information about the line, such as its slope and where it intersects the y-axis.
Linear equations translate real-world relationships into mathematical language. For example, if you are calculating the cost over time with a repeating charge, a linear equation can neatly express that relationship. In our equation, \( f(x) = 3x + 12 \), it captures both the rate of change (3) and the initial value (12), creating a full picture of the function's behavior on a graph.

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Most popular questions from this chapter

Are the lines parallel? $$ y=2+3(x+5) ; y=2+4(x+5) $$

Write an expression in \(x\) representing the result of the given operations on \(x\). Is the expression linear in \(x ?\) Add \(5,\) multiply by \(2,\) subtract \(x\).

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Which equation, (a)-(d), has the graph that crosses the \(y\) -axis at the highest point? (a) \(y=3(x-1)+5\) (b) \(x=3 y+2\) (c) \(y=1-6 x\) (d) \(2 y=3 x+1\)
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