Problem 11 - Precalculus with Limits

Question

In Exercises 7-14, determine whether each point lies on the graph of the equation. \( y = |x - 1| + 2 \) (a) \( (2, 3) \) (b) \( (-1, 0) \)

Step-by-Step Solution

Answer

(a) Point \( (2, 3) \) lies on the graph. (b) Point \(-1, 0\) does not lie on the graph.

Step 1: Substitute Point A into the Function

For point A \((2,3)\), substitute 2 for x and 3 for y in the function \( y = |x - 1| + 2 \). This gives: \( 3 = |2 - 1| + 2 \). Simplify to check if the left-hand side equals the right-hand side.

Step 2: Verify Equality for Point A

The absolute value of \(2 - 1\) is 1, so the right-hand side of the equation becomes \( 1 + 2 = 3 \), which equals the y-coordinate of point A. Thus, point A lies on the graph.

Step 3: Substitute Point B into the Function

For point B \(-1, 0)\), substitute -1 for x and 0 for y in the function \( y = |x - 1| + 2 \). This gives: \( 0 =|-1 - 1| + 2 \). Simplify to check if the left-hand side equals the right-hand side.

Step 4: Verify Equality for Point B

The absolute value of \(-1 - 1\) is 2, so the right-hand side of the equation becomes \( 2 + 2 = 4 \), which does not equal the y-coordinate of point B. Thus, point B does not lie on the graph.

Key Concepts

Graph of an Equation

The graph of an equation represents the set of all points that satisfy the given equation. Understanding how to graph an equation helps us visualize relationships between variables. In this exercise, we are dealing with an absolute value function: \( y = |x - 1| + 2 \).

Absolute value functions form a V-shape on the coordinate plane. The vertex of this graph is crucial and occurs at the point where the expression inside the absolute value equals zero. Here, this happens when \( x = 1 \). Thus, at \( x = 1 \), the value of the function is the smallest possible, which is 2, resulting in the vertex at \( (1, 2) \).

When graphing, it's important to identify key points, such as the vertex and intercepts, and understand how the absolute value impacts the shape. The graph will rise away from the vertex in both directions. This is due to the nature of absolute values always yielding non-negative results.

When solving these types of problems, the graph shows visually which points lie on it, providing an intuitive understanding beyond numerical calculations.

Substitution Method

The substitution method is a straightforward approach used to determine if a particular point lies on the graph of an equation.

For the equation \( y = |x - 1| + 2 \), you substitute the x-value of the point into the equation and calculate the resulting y-value. Then, you compare this calculated y-value with the y-coordinate of the point you're testing.

Here's a step-by-step approach:

• Take point (2, 3): Substitute \( x = 2 \), compute \( y = |2 - 1| + 2 \), simplify to find \( y = 3 \).
• Since the computed y-value matches the y-coordinate (3), Point A (2, 3) lies on the graph.
• For Point B (-1, 0): Substitute \( x = -1 \), leading to \( y = |-1 - 1| + 2 \). Simplify to find \( y = 4 \).
• Since the calculated y-value (4) does not match the y-coordinate (0), Point B does not lie on the graph.

By testing the point through substitution, you directly check whether it satisfies the original equation, hence confirming its presence on the graph.

Verifying Points on a Graph

Verifying if points lie on a graph is essential to ensure the accuracy of solutions and understanding.

The process involves calculating using the given equation and comparing specific coordinates of the points involved.

For example, the point (2, 3) was verified to be on the graph of the equation \( y = |x - 1| + 2 \) by confirming that substituting \( x = 2 \) results in \( y = 3 \).

To verify points:

• Substitute the x-value into the equation to generate a y-value.
• Compare this generated y-value with the y-coordinate of the point.
• If they match, the point lies on the graph; if not, it doesn't.

This strategy is beneficial for any equation type, especially when dealing with absolute value expressions that can seem tricky at first. Verifying points also deepens your understanding of the graph's behavior and confirms correct calculations. Try practicing this with different points to become familiar with verifying points on a graph accurately.