Ch. 2 - Graphs and Functions

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

Lial 13th Edition

Ch. 2 - Graphs and Functions

Problem 3

Chapter 3, Problem 3

Find the distance between each pair of points, and give the coordinates of the midpoint of the line segment joining them. A(-6, 3), B(-6,8)

Verified step by step guidance

Identify the coordinates of the two points: A(-6, 3) and B(-6, 8).

Use the distance formula to find the distance between points A and B: $\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$, where $(x_1, y_1)$ and $(x_2, y_2)$ are the coordinates of points A and B respectively.

Substitute the coordinates into the distance formula: $\sqrt{(-6 - (-6))^2 + (8 - 3)^2}$.

Use the midpoint formula to find the midpoint coordinates: $\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)$.

Substitute the coordinates into the midpoint formula: $\left( \frac{-6 + (-6)}{2}, \frac{3 + 8}{2} \right)$.

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Key Concepts

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

Distance Formula

The distance formula calculates the length between two points in the coordinate plane. It is derived from the Pythagorean theorem and given by d = √[(x2 - x1)² + (y2 - y1)²], where (x1, y1) and (x2, y2) are the coordinates of the points.

Midpoint Formula

The midpoint formula finds the point exactly halfway between two given points. It is calculated as ((x1 + x2)/2, (y1 + y2)/2), providing the average of the x-coordinates and y-coordinates of the endpoints.

Coordinate Geometry Basics

Understanding the Cartesian coordinate system is essential, where points are represented as (x, y). This knowledge helps in visualizing points, line segments, and applying formulas like distance and midpoint effectively.

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Without using paper and pencil, evaluate each expression given the following functions. $ƒ (x) = x + 1$ and $g (x) = x^{2}$

$open paren f with hook slash g close paren times 2$

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Fill in the blank to correctly complete each sentence. The point (4,_____ ) lies on the graph of the equation y = 3x - 6.

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Fill in the blank(s) to correctly complete each sentence. The circle with center (3, 6) and radius 4 has equation _________.

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Fill in the blank(s) to correctly complete each sentence. The equation y = 4x - 6 defines a function with independent variable______ and dependent variable ________ .

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Without using paper and pencil, evaluate each expression given the following functions. $ƒ (x) = x + 1$ and $g (x) = x^{2}$

(ƒg)(2)

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Textbook Question

To answer each question, refer to the following basic graphs. Which one is the graph of ƒ(x)=x³? What is its range?

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