Problem 48 - Algebra 1: Concepts and Skills

Question

Graph the function. $$ h(x)=-3 x-1 $$

Step-by-Step Solution

Answer

The graph of the function \( h(x) = -3x -1 \) is a straight line passing through the points (0, -1) and (1, -4), with a slope of -3 and y-intercept at -1.

Step 1: Identify the slope and y-intercept

From the given equation \( h(x) = -3x - 1 \), the number in front of \( x \), which is -3, is the slope \( m \), and the constant term -1 is the y-intercept \( b \). Thus, the slope is -3 and the y-intercept is -1.

Step 2: Plot the y-intercept

The y-intercept is the point at which the line crosses the y-axis. Given that the y-intercept is -1, this point on the graph is (0, -1). Mark this point on the graph.

Step 3: Use the slope to find another point

The slope is the rise over run, or the change in y divided by the change in x. Here the slope is -3, meaning for every unit moved to the right on the x-axis, we move 3 units down on the y-axis. Starting from the y-intercept (0, -1), move 1 unit to the right and 3 units down to find the point (1, -4). Mark this point on the graph.

Step 4: Draw the line

Draw a straight line passing through the points (0, -1) and (1, -4). This line represents the graph of the function \( h(x) = -3x - 1 \).

Key Concepts

Slope

The slope is a key concept in understanding linear functions. It represents the steepness and direction of a line on a graph. In every linear equation of the form \( y = mx + b \), the letter \( m \) stands for the slope. The slope is calculated as "rise over run," which means the "change in \( y \)" divided by the "change in \( x \)."
For example, in the equation \( h(x) = -3x - 1 \), the slope is \( -3 \). This negative value indicates that the line is slanting downwards from left to right on the coordinate plane. A slope of \( -3 \) tells us that for every 1 unit you move to the right along the x-axis, you go down 3 units on the y-axis.

Key points about slope include:

• A positive slope means the line rises as it moves from left to right.
• A negative slope means the line falls as it moves from left to right.
• A slope of zero indicates a horizontal line.
• An undefined slope indicates a vertical line.

Y-intercept

The y-intercept of a linear function is the point where the line crosses the y-axis on a graph. It is an essential component of the equation because it tells you where the line starts on the y-axis.
In the equation \( h(x) = -3x - 1 \), the y-intercept is \( -1 \). This means the first point on the graph where the line intersects the y-axis is at \( (0, -1) \).

The y-intercept has these important qualities:

• The y-intercept is always at \( x = 0 \).
• It can be found directly from the linear equation as the constant term.
• It provides a specific point that helps in graphing the line.

The y-intercept is a fixed starting point on the graph, from which you can use the slope to find additional points to draw the line.

Linear Equation

A linear equation is an algebraic equation that forms a straight line when graphed on a coordinate plane. The general form of a linear equation is \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
In a linear equation, the relationship between \( x \) and \( y \) is constant, meaning the line has only one fixed slope. Linear equations are simple yet powerful tools for modeling real-world situations, predict trends, and determine relationships between variables.

Some key points about linear equations:

• The graph of a linear equation is always a straight line.
• Linear equations can represent growth and decay by altering the slope.
• They are used in various fields, including economics, physics, and statistics.
• Understanding linear equations helps solve various mathematical problems effectively.

Knowing how to graph and interpret linear equations is a foundational skill in algebra and calculus.

Coordinate Plane

The coordinate plane is a two-dimensional surface where you can plot points, lines, and curves using a pair of numerical values. It consists of two intersecting perpendicular lines, known as the x-axis (horizontal) and the y-axis (vertical).
Points on the coordinate plane are defined by ordered pairs \( (x, y) \), where \( x \) is the horizontal value, and \( y \) is the vertical value. For instance, the point \( (0, -1) \) is found at zero units on the x-axis and negative one unit on the y-axis.

Key features of the coordinate plane include:

• The point \( (0, 0) \) is called the origin, where the x-axis and y-axis intersect.
• It is divided into four quadrants, each representing different signs of x and y values.
• It allows the visualization of linear functions and other graphs.

Understanding the coordinate plane is essential for graphing equations, analyzing functions, and comprehending geometric concepts.