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Chapter 6: Problem 1

State the gradient and vertical intercept of (a) \(y=8 x-3\) (b) \(y=3 t-2\) (c) \(y=9\) (d) \(y=-3 t\) (e) \(f(x)=-3-4 x\)

Short Answer

Expert verified
(a) Gradient: 8, Intercept: -3; (b) Gradient: 3, Intercept: -2; (c) Gradient: 0, Intercept: 9; (d) Gradient: -3, Intercept: 0; (e) Gradient: -4, Intercept: -3.

Step by step solution

01

Understanding the Form of the Equation

The general form of a linear equation is \( y = mx + c \), where \( m \) is the gradient, and \( c \) is the vertical intercept of the line. This pattern will help us identify the values for each part.
02

Analyzing Equation (a) \( y = 8x - 3 \)

Comparing \( y = 8x - 3 \) to the general form, we identify \( m = 8 \) and \( c = -3 \). Thus, the gradient is 8, and the vertical intercept is -3.
03

Analyzing Equation (b) \( y = 3t - 2 \)

Treating \( t \) as the same as \( x \) in the form, compare \( y = 3t - 2 \) to \( y = mx + c \). Here, \( m = 3 \) and \( c = -2 \). The gradient is 3, and the vertical intercept is -2.
04

Analyzing Equation (c) \( y = 9 \)

This equation is a horizontal line, thus \( m = 0 \) because there is no \( x \) term. The vertical intercept \( c \) is 9. The gradient is 0, and the vertical intercept is 9.
05

Analyzing Equation (d) \( y = -3t \)

Here, compare \( y = -3t \) to the form \( y = mx + c \). We see \( m = -3 \) and \( c = 0 \) as there is no additional constant. The gradient is -3, and the vertical intercept is 0.
06

Analyzing Equation (e) \( f(x) = -3 - 4x \)

Rearranging it to \( y = -4x - 3 \), compare with \( y = mx + c \). This gives \( m = -4 \) and \( c = -3 \). The gradient is -4, and the vertical intercept is -3.

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Key Concepts

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

Gradient
The gradient of a line, often known as the slope, describes how steep the line is. It is represented by the letter \( m \) in the equation of a line in the slope-intercept form, \( y = mx + c \). The gradient tells us how much \( y \) changes for a unit change in \( x \).
Here鈥檚 how to understand it:
  • A positive gradient means the line rises as it moves from left to right.
  • A negative gradient means the line falls as you move along the horizontal axis.
  • A gradient of zero indicates a perfectly horizontal line, where \( y \) does not change regardless of the \( x \) value.
When given a linear equation, the coefficient of \( x \) is your gradient. So, for the equation \( y = 8x - 3 \), the gradient is \( m = 8 \). This tells you that for every step you take to the right along the \( x \)-axis, the line goes up by 8 units.
Vertical Intercept
The vertical intercept, also known as the \( y \)-intercept, is the point where the line crosses the \( y \)-axis. In the slope-intercept form, \( y = mx + c \), the \( c \) represents the vertical intercept. This is the value of \( y \) when \( x \) equals 0.
To identify the vertical intercept, look for the constant term in the equation. Some key points include:
  • For the equation \( y = 8x - 3 \), the vertical intercept is \( -3 \). This means that when \( x \) is 0, \( y \) is \(-3\).
  • If there is no constant term, it means the vertical intercept is \( 0 \), and the line passes through the origin. For example, in \( y = -3t \), the intercept is \( 0 \).
Understanding the vertical intercept helps to plot the line and grasp the initial value of \( y \) before any change in \( x \).
Linear Functions
Linear functions are foundational in algebra and are represented by straight lines on a graph. They show a constant rate of change, making them predictable and easy to understand. The general form of a linear equation is \( y = mx + c \).
Characteristics of linear functions include:
  • They form a straight line when plotted on a graph.
  • The rate of change is constant across all values of \( x \); this is the gradient.
  • The behavior of the line either increases, decreases, or remains constant depending on the gradient.
Linear functions are ubiquitous in real-life scenarios, such as calculating travel distance over time or predicting profits from sales. They provide a simple model for understanding how changes in one variable affect another.
Slope-Intercept Form
The slope-intercept form is one of the most common ways of expressing a linear equation. It is written as \( y = mx + c \), where:
  • \( m \) represents the slope or gradient of the line.
  • \( c \) is the \( y \)-intercept, where the line crosses the \( y \)-axis.
Using this form makes it easy to identify the key characteristics of the line, such as its steepness and starting point. For example, in \( y = 3t - 2 \), the slope-intercept form tells us that the line's gradient is \( 3 \) and it crosses the \( y \)-axis at \( -2 \).
Here鈥檚 why it鈥檚 useful:
  • Quickly determine how the output \( y \) changes based on different inputs \( x \).
  • Helps in graphing lines with ease by knowing just the slope and where to start on the \( y \)-axis.
  • Facilitates easier comparison between different linear equations.
Employ the slope-intercept form to easily sketch graphs and analyze the relationship between variables in both academic and practical applications.

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