Problem 49 - College Algebra

Question

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

Step-by-Step Solution

Answer

The translated mathematical equation is \(y = 3 - x^2\). On graphing this equation, a downward-opening parabola is obtained. The minimum value of \(y\) is taken at \(x = 0\) where the \(y\)-value is 3, and it decreases as \(x\) deviates from zero.

Step 1: Translate English Sentence Into a Mathematical Equation

The English sentence 'The \(y\)-value is three decreased by the square of the \(x\)-value' can be translated into an equation. 'Three decreased by' is translated as subtraction from three, and 'the square of the \(x\)-value' is \(x^2\), thus the equation becomes: \(y = 3 - x^2\).

Step 2: Define Range for \(x\)

To graph the above equation, we need to determine the range of \(x\). Normally we select a range like \(-3 \leq x \leq 3\) to include positive and negative \(x\) values.

Step 3: Create a Table of Values

We create a table of values for \(x\) and corresponding \(y\) values by substituting the \(x\) values (-3, -2, -1, 0, 1, 2, 3) in the equation. Calculate the \(y\) value for each.

Step 4: Sketch the Graph

Using the table of values, plot the coordinates \((x, y)\) on a graph and connect the points to draw the curve of the equation \(y = 3 - x^2\). The curve will be a downward-opening parabola.