Problem 8 - Algebra and Trigonometry Real Mathematics, Real People

Question

Determine whether each point lies on the graph of the equation. \(y=|x-1|+2\) (a) (2,1) (b) (3.2,4.2)

Step-by-Step Solution

Answer

The point (2,1) does not lie on the graph while the point (3.2,4.2) does lie on the graph of the equation \(y=|x-1|+2\).

Step 1: Substituting (2,1) into the equation

Put \(x=2\) and \(y=1\) into the equation \(y=|x-1|+2\). It will look like this: 1=|2-1|+2 which simplifies to 1=1+2. Since \(1 \ne 3\), the point (2,1) does not lie on the graph.

Step 2: Substituting (3.2,4.2) into the equation

Let's substitute \(x = 3.2\) and \(y = 4.2\) into the equation \(y=|x-1|+2\). That leads to 4.2=|3.2-1|+2 which simplifies to 4.2=2.2+2. Since \(4.2 = 4.2\), that holds true. Therefore, the point (3.2,4.2) does lie on the graph.

Key Concepts

Solving Equations

The process of solving equations involves finding the variable values that make the equation true. In this exercise, we substitute points into an absolute value function to determine if they lie on its graph. The equation given is \[ y = |x - 1| + 2 \]. To solve, we replace \(x\) and \(y\) with the coordinates of our point. If the equation holds true, meaning both sides are equal, the point lies on the graph.

• Consider the point (2, 1): Plug these values into the equation and see if it results in a true statement. This substitution gives us \(1 = |2 - 1| + 2\). After simplifying, we have \(1 = 3\), which is false. Therefore, (2, 1) does not satisfy the equation.
• Now try (3.2, 4.2): Substitute to get \(4.2 = |3.2 - 1| + 2\), which simplifies to \(4.2 = 4.2\). This is true, indicating that (3.2, 4.2) satisfies the equation and is indeed on the graph.

Remember, solving such equations involves verifying that substitutions make the equation equal on both sides.

Coordinate Geometry

Coordinate geometry is a fascinating branch of mathematics where algebra meets geometry. It deals with graphing equations such as lines, curves, and in this case, absolute value functions. When graphing \( y = |x - 1| + 2 \), we look for how figures and shapes, like points, relate spatially on the coordinate plane.The key aspects of coordinate geometry you'll consider are:

• X and Y-Axes: These help locate points (x, y) on the plane. The x-axis runs horizontally, and the y-axis runs vertically.
• Distance from a Point: Absolute value in our equation \(|x-1|\) reflects how far a number is from zero on a number line, which pivots around the x-coordinate of the vertex; you adjust with the +2 vertically indicating the shift up or down from the line.
• Graphing: When plotting, the graph opens in a V-shape, starting from the vertex at (1, 2) (direction determined by coefficient/sign in front of x). Points you substitute help show if they lie within this V's arms.

Understanding these components makes graph interpreting intuitive as you verify whether specific coordinates are true reflections on the chart.

Mathematical Reasoning

Mathematical reasoning is about logical thinking and deduction. Applied here, it helps determine whether points lie on a graph by evaluating the truth of equations. This type of reasoning requires you to logically follow through with substitution and simplification steps in the given function.

• Substitution Method: By placing point values into the function, you check if they satisfy the relation, thus lying on the graph.
• Simplification Process: Often involves reducing expressions, especially when handling absolute values, to verify equation equality. For instance, substitute known values and reduce, as illustrated with \(1 eq 3\), helping conclude the point's irrelevance to the graph.
• Logical Deduction: Upon completing calculations, deductions assist in confirming or denying point placement. If \(4.2 = 4.2\), the initial assertion holds, and point validation completes.

By applying these reasoning skills, mathematicians confirm geometric truths in abstract algebraic contexts, moving from problem statements to coherent conclusions.