Problem 1 - Algebra A Combined Function

Question

Use the slope-intercept form to graph each equation. $$ y=2 x+1 $$

Step-by-Step Solution

Answer

Plot the y-intercept (0,1), use slope 2 to find another point (1,3), then draw the line through these points.

Step 1: Understand the Slope-Intercept Form

The slope-intercept form of a line equation is given by \( y = mx + b \), where \( m \) represents the slope of the line and \( b \) represents the y-intercept, the point where the line crosses the y-axis.

Step 2: Identify Slope and Y-Intercept

From the given equation \( y = 2x + 1 \), identify the slope \( m = 2 \) and the y-intercept \( b = 1 \). This means the line crosses the y-axis at \((0,1)\).

Step 3: Plot the Y-Intercept

Start by plotting the y-intercept on the graph. The y-intercept is the point \((0,1)\). Mark this point on the graph.

Step 4: Use the Slope to Find Another Point

The slope \( m = 2 \) means rise over run is \( \frac{2}{1} \). Starting at the y-intercept \((0,1)\), move up 2 units and then 1 unit to the right to reach the new point \((1,3)\).

Step 5: Draw the Line

Using a ruler, draw a straight line through the points \((0,1)\) and \((1,3)\). Extend the line across the graph grid to complete the graph of the equation \( y = 2x + 1 \).

Key Concepts

Graphing Linear Equations

Graphing linear equations is a fundamental skill in mathematics, allowing you to visualize relationships between variables. A linear equation in two variables, such as \( y = 2x + 1 \), represents a straight line when graphed on a coordinate plane. This process involves understanding the equation's form and using it to plot the line.

When graphing, it helps to start by identifying the slope-intercept form, which is \( y = mx + b \). This form makes it easier to determine crucial points, like where the line will cross the y-axis. Equipped with this information, you can accurately sketch the line on a graph.

Remember, graphing isn't just about plotting points; it helps to clearly understand the relationship represented by the equation. Linear graphs can quickly show how one variable changes in relation to another, providing a visual way to interpret equations.

Slope and Y-Intercept

The slope and y-intercept are key components of the slope-intercept form, helping you understand the line's direction and position. For the equation \( y = 2x + 1 \), the slope is represented by \( m = 2 \), and the y-intercept by \( b = 1 \).

• The slope, \( m \), reflects how steep the line is. A positive slope, such as 2, indicates an upward slant from left to right. This tells us that as \( x \) increases by 1, \( y \) increases by 2.
• The y-intercept, \( b \), is where the line crosses the y-axis, at the point \((0, b)\). Here, it's at \((0, 1)\), meaning if \( x = 0 \), then \( y = 1 \).

Understanding these elements helps you confidently graph the equation by providing the starting point and direction for your line.

Plotting Points

Plotting points is a critical step in graphing any equation. It involves placing specific points onto a coordinate plane, which will define the shape and position of the graph. For a linear equation like \( y = 2x + 1 \), you begin by plotting the y-intercept, which is straightforward.

Start at the point \((0,1)\), where the line crosses the y-axis. Use the slope to determine the next point. The slope of 2 means that for every 1 unit you move to the right, you move up 2 units. From \((0,1)\), moving right 1 unit and up 2 units places you at \((1,3)\).

By connecting \((0,1)\) and \((1,3)\) with a straight line, you establish the path that extends through those points, completing the graph. Successful plotting ensures the graph accurately represents the equation, showcasing how one variable affects the other.