Problem 89 Graph each equation by completin... [FREE SOLUTION]

91影视

Beginning Algebra

Margaret L. Lial, John Hornsby

$Math Studyset 91影视 Explanations$ Math

11 Edition

Chapter 5: Problem 89

Graph each equation by completing the table of values. $y=x^{2}-4$ $\begin{array}{|c|c|}\hline x & {y} \\ \hline-2 & {} \\ \hline-1 & {} \\\ \hline 0 & {} \\ \hline 1 & {} \\ \hline 2 & {} \\ \hline\end{array}$

Short Answer

Expert verified

The points to graph are $(-2, 0), (-1, -3), (0, -4), (1, -3), (2, 0)$.

Step by step solution

Understand the given equation

The equation to graph is given as $ y = x^2 - 4 $. This is a quadratic equation, meaning it will form a parabola when graphed.

Create the table of values

Complete the given table by substituting each value of $ x $ into the equation $ y = x^2 - 4 $ and solving for $ y $.

Calculate $ y $ for $ x = -2 $

Substitute $ x = -2 $ into the equation: $ y = (-2)^2 - 4 = 4 - 4 = 0 $. Fill this value into the table.

Calculate $ y $ for $ x = -1 $

Substitute $ x = -1 $ into the equation: $ y = (-1)^2 - 4 = 1 - 4 = -3 $. Fill this value into the table.

Calculate $ y $ for $ x = 0 $

Substitute $ x = 0 $ into the equation: $ y = 0^2 - 4 = 0 - 4 = -4 $. Fill this value into the table.

Calculate $ y $ for $ x = 1 $

Substitute $ x = 1 $ into the equation: $ y = 1^2 - 4 = 1 - 4 = -3 $. Fill this value into the table.

Calculate $ y $ for $ x = 2 $

Substitute $ x = 2 $ into the equation: $ y = 2^2 - 4 = 4 - 4 = 0 $. Fill this value into the table.

Plot the points and draw the graph

Use the completed table of values to plot the points $(-2, 0), (-1, -3), (0, -4), (1, -3), (2, 0)$ on a graph. Then, connect the points to form a parabola.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

parabola

A quadratic equation, such as the given equation $ y = x^2 - 4 $, forms a special type of graph called a parabola. A parabola is a symmetrical, U-shaped curve that can open either upwards or downwards.
In our equation, the parabola opens upwards because the coefficient of $ x^2 $ is positive. The vertex of the parabola, which is either the highest or lowest point, is the turning point of the curve. For $ y = x^2 - 4 $, the vertex is at the point $ (0, -4) $.
Understanding the shape and orientation of a parabola helps in sketching the graph accurately.

table of values

To graph the equation $ y = x^2 - 4 $, we start by creating a table of values.
This table assists in finding the precise points on the graph. Here鈥檚 how to approach it:

Choose a range of $ x $ values, usually covering both negative and positive numbers for symmetry.
Substitute each value of $ x $ into the equation to calculate $ y $.
Fill these values into the table to get coordinate pairs $ (x, y) $.

A complete table for this problem is:
For $ x = -2 $, $ y = 0 $;
For $ x = -1 $, $ y = -3 $;
For $ x = 0 $, $ y = -4 $;
For $ x = 1 $, $ y = -3 $;
For $ x = 2 $, $ y = 0 $.
This table provides the necessary points to plot on the graph.

substitution method

The substitution method involves replacing the variable $ x $ in the equation with specific values to determine $ y $. Here鈥檚 a step-by-step rundown with our example:

For $ x = -2 $: $ y = (-2)^2 - 4 = 4 - 4 = 0 $.
For $ x = -1 $: $ y = (-1)^2 - 4 = 1 - 4 = -3 $.
For $ x = 0 $: $ y = 0^2 - 4 = 0 - 4 = -4 $.
For $ x = 1 $: $ y = 1^2 - 4 = 1 - 4 = -3 $.
For $ x = 2 $: $ y = 2^2 - 4 = 4 - 4 = 0 $.

Each substitution computes a corresponding $ y $ value, forming coordinate pairs like $ (-2, 0) $, $ (-1, -3) $, $ (0, -4) $, etc.

plotting points on graph

After calculating the coordinates, it's time to plot these points on a graph. Use a Cartesian plane where the $ x $-axis is horizontal and the $ y $-axis is vertical.
Follow these steps:

Mark each coordinate pair on the plane.
For $ (-2, 0) $, move 2 units left of the origin and mark 0 units up.
For $ (-1, -3) $, move 1 unit left and 3 units down.
Continue for the rest: $ (0, -4) $, $ (1, -3) $, $ (2, 0) $.
Connect these points smoothly to reveal the upward-opening parabola.

This process shows the beautiful, symmetrical shape of the quadratic function.

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91影视

Graph each equation by completing the table of values. \(y=x^{2}-4\) \(\begin{array}{|c|c|}\hline x & {y} \\ \hline-2 & {} \\ \hline-1 & {} \\\ \hline 0 & {} \\ \hline 1 & {} \\ \hline 2 & {} \\ \hline\end{array}\)

Short Answer

Step by step solution

Understand the given equation

Create the table of values

Calculate \( y \) for \( x = -2 \)

Calculate \( y \) for \( x = -1 \)

Calculate \( y \) for \( x = 0 \)

Calculate \( y \) for \( x = 1 \)

Calculate \( y \) for \( x = 2 \)

Plot the points and draw the graph

Key Concepts

parabola

table of values

substitution method

plotting points on graph

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