Ch. 2 - Graphs and Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 9

Chapter 3, Problem 9

Determine whether each statement is true or false. If false, explain why. The midpoint of the segment joining (0, 0) and (4, 4) is 2.

Verified step by step guidance

Recall the formula for the midpoint of a segment joining two points $(x_1, y_1)$ and $(x_2, y_2)$: \[\text{Midpoint} = \left( \frac{\,x_1 + x_2}{2}, \frac{\,y_1 + y_2}{2} \right)\]

Identify the given points: $(0, 0)$ and $(4, 4)$. Substitute these into the midpoint formula: \[\left( \frac{0 + 4}{2}, \frac{0 + 4}{2} \right)\]

Simplify each coordinate: \[\left( \frac{4}{2}, \frac{4}{2} \right) = (2, 2)\]

Analyze the statement: it claims the midpoint is 2, which is a single number, not a point with two coordinates. Since the midpoint is $(2, 2)$, the statement is false.

Explain why: The midpoint of a segment in the coordinate plane is always a point with two coordinates, not a single number. Therefore, saying the midpoint is 2 is incorrect.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Midpoint Formula

The midpoint of a line segment between two points (x₁, y₁) and (x₂, y₂) is found by averaging their coordinates: ((x₁ + x₂)/2, (y₁ + y₂)/2). This gives the exact center point on the segment.

Coordinate Points in the Plane

Points in the coordinate plane are represented as ordered pairs (x, y). Understanding how to interpret and manipulate these pairs is essential for calculating distances, midpoints, and other geometric properties.

Interpreting Statements About Points

Statements about points must be precise; for example, a midpoint is a point with two coordinates, not a single number. Recognizing the difference between a point and a scalar value is crucial for evaluating the truth of such statements.

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Related Practice

Textbook Question

Determine whether each statement is true or false. If false, explain why. The graph of y = x² + 2 has no x-intercepts.

Textbook Question

Fill in the blank(s) to correctly complete each sentence. The function $g (x) = √ x$ has domain ________.

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Textbook Question

In the following exercises, (a) find the center-radius form of the equation of each circle described, and (b) graph it. center (0, 0), radius 6

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Textbook Question

Let ƒ(x)=x²+3 and g(x)=-2x+6. Find each of the following. (ƒ+g)(3)

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Textbook Question

To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=√x? What is its domain?

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Textbook Question

Without actually graphing, identify the type of graph that each equation has.

$x^2+y^2=144$

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