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Chapter 1: Problem 50

Write each English sentence as an equation in two variables. Then graph the equation. The \(y\) -value is two more than the square of the \(x\) -value.

Short Answer

Expert verified
The equation is \(y = x^2 + 2\). The graph is a parabola opening upwards with its vertex at (0, 2).

Step by step solution

01

Translate the sentence into an equation

The sentence tells us that \(y\) (the 'y-value') is equal to the square of \(x\) (which we can write as \(x^2\)) plus two. We can write this relationship as an equation: \(y = x^2 + 2\).
02

Create a table of values

To graph the equation, it can be helpful to first create a table of values. This involves choosing a few values for \(x\) (for example, -3, -2, -1, 0, 1, 2, 3) and calculating the corresponding \(y\)-values using the equation \(y = x^2 + 2\). The pairs \((x, y)\) will be the coordinates of points on the graph.
03

Plot the values on a graph

Plot the points from the table of values on a graph with \(x\) on the horizontal axis and \(y\) on the vertical axis. The equation \(y = x^2 + 2\) is a parabola, so the points should form a U-shaped curve.
04

Draw the graph

Connect the plotted points with a smooth curve to complete the graph of the equation. Make sure that the curve gets wider as it leaves the vertex, and never crosses the line \(y = 2\), because the \(y\)-values are always greater than or equal to 2 (since they're calculated by squaring \(x\) and adding 2, and squaring any real number gives a nonnegative result).

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