Problem 83 - Algebra 1: Concepts and Skills

Question

Use a table of values to graph the equation. \(y=\frac{3}{4} x+2\)

Step-by-Step Solution

Answer

The linear graph will pass through the points (-2,-1.5), (0,2) and (2,3.5).

Step 1: Build the Table

We select values for x as -2, 0, and 2 and substitute these into \(y=\frac{3}{4} x+2\) to find the corresponding y values.

Step 2: Substitute x Values

For x=-2, y become \(y=\frac{3}{4}*(-2)+2 = -1.5\), for x=0, y becomes \(y=\frac{3}{4}*0+2 = 2\), and for x=2, y becomes \(y=\frac{3}{4}*2+2 = 3.5\).

Step 3: Plot the Coordinates

On the graph, plot the coordinate pairs (-2,-1.5), (0,2) and (2,3.5).

Step 4: Draw the Line

Join these three points with a straight line. Ensure that the line extends on both ends beyond the points plotted. This ensures the line for all possible values of x.

Key Concepts

Graphing

Graphing linear equations is a fundamental skill in algebra, visualizing relationships between variables. Consider the equation \( y=\frac{3}{4} x+2 \). This represents a straight line on a graph.

The highest exponent of \(x\) in this equation is 1, indicating it's a linear equation. To "graph" it, you must plot points and connect them with a straight line.

• First, identify the slope (\(\frac{3}{4}\)) and y-intercept (2). These tell you the steepness and where the line crosses the y-axis.
• Understand that the graph represents all solutions of the equation, forming an endless line through those points.
• Remember to extend the line infinitely in both directions, as the real number solutions continue beyond just your plotted points.

Reading the graph helps predict other values of \(y\) for any value of \(x\). A clear graph can visually convey data and relationships.

Table of Values

Using a table of values is an essential step in graphing linear equations. This method simplifies finding points to plot on a graph. For our equation \( y=\frac{3}{4} x+2 \), here's how to proceed:

• Choose any numbers for \(x\) (e.g., -2, 0, 2), forming the basis of your table.
• Plug these \(x\) values into the equation to compute corresponding \(y\) values.
• Record these \(x, y\) pairs in your table. This gives coordinate points like (-2, -1.5), (0, 2), and (2, 3.5).

This table becomes a guide to plot points accurately on a coordinate plane. Each entry links \(x\) to its paired \(y\), simplifying the process of forming the line of an equation.

Coordinate Plane

The coordinate plane is where we visualize equations like \( y=\frac{3}{4} x+2 \). It's a flat surface defined by a horizontal x-axis and a vertical y-axis intersecting at the origin (0,0).

When working with a coordinate plane:

• Each point on the plane is identified by a pair of numbers (x, y), marking its horizontal and vertical positioning.
• The plane allows you to graphically represent algebraic equations, making them easier to interpret.
• It helps observe how changes in \(x\) affect \(y\), showcasing concepts like slope visually.

By plotting the coordinates from the table of values, you can draw the equation as a visible straight line on the coordinate plane, highlighting relationships among variables.