Problem 3 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-20: Sketch the graph of the equation, and label the \(x\) - and \(y\)-intercepts. $$ y=-x+1 $$

Step-by-Step Solution

Answer

Graph the line through points (0, 1) and (1, 0), labeling intercepts.

Step 1: Understanding the Equation

The given equation is a linear equation in the slope-intercept form, which is \( y = mx + c \). In this equation, \( m \) is the slope and \( c \) is the y-intercept. Here, the equation is \( y = -x + 1 \), so the slope \( m = -1 \) and the y-intercept \( c = 1 \).

Step 2: Find the Y-Intercept

To find the y-intercept, set \( x = 0 \) in the equation. This gives us \( y = -0 + 1 \), so \( y = 1 \). The y-intercept is the point where the graph crosses the y-axis, which is at the point \( (0, 1) \).

Step 3: Find the X-Intercept

To find the x-intercept, set \( y = 0 \) in the equation. This gives us \( 0 = -x + 1 \). Solving for \( x \), we get \( x = 1 \). Therefore, the x-intercept is the point where the graph crosses the x-axis, which is at the point \( (1, 0) \).

Step 4: Plot the Intercepts and Graph the Line

Plot the y-intercept \( (0, 1) \) and the x-intercept \( (1, 0) \) on a coordinate plane. Draw a straight line through these two points. This line represents the graph of the equation \( y = -x + 1 \).

Step 5: Label the Intercepts

On the graph, label the point \( (0, 1) \) as the y-intercept and the point \( (1, 0) \) as the x-intercept. This helps in identifying these key points on the line.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line so that it can be easily graphed or understood. This form is written as \( y = mx + c \).Each component of this equation has a significant role:

• \( y \) is the dependent variable, which depends on \( x \).
• \( m \) represents the slope of the line. The slope shows how steep the line is, or how quickly \( y \) changes as \( x \) changes.
• \( x \) is the independent variable.
• \( c \) is the y-intercept, representing where the line crosses the y-axis.

In our equation \( y = -x + 1 \), the slope \( m = -1 \). This means that for every one unit increase in \( x \), \( y \) decreases by one unit. The negative slope indicates that the line is slanting downwards from left to right. Adjusting the slope and intercept independently allows you to graph many different lines quickly and accurately.

X-Intercept

The x-intercept of a line is a point where it crosses the x-axis. At this point, the value of \( y \) is zero. To find the x-intercept, set \( y = 0 \) in the equation and solve for \( x \).Let's break down an example using the equation \( y = -x + 1 \):

• Set \( y = 0 \), leading to the equation \( 0 = -x + 1 \).
• Solving for \( x \), you'll subtract 1 from both sides and then divide by -1: \( x = 1 \).
• The x-intercept is \( (1, 0) \).

On a graph, this point will be where the line crosses the x-axis. The x-intercept gives a tangible value of \( x \) when \( y \) equals zero, which is vital for understanding the behavior of the line.

Y-Intercept

The y-intercept is a crucial part of understanding the behavior of a linear equation. It is the point where the line crosses the y-axis, and it occurs when \( x = 0 \). You can easily find the y-intercept by setting \( x \) to zero in the equation.Here's how it works in the equation \( y = -x + 1 \):

• Set \( x = 0 \), substituting into the equation yields \( y = -0 + 1 \).
• This simplifies to \( y = 1 \).
• Thus, the y-intercept is the point \( (0,1) \).

This point is very helpful when drawing the graph because it gives a starting point from which you can apply the slope to find other points on the line. Recognizing the y-intercept helps distinguish how far up or down the line is from the origin when \( x \) equals zero.