Problem 50 - College Algebra Essentials

Question

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.

Step-by-Step Solution

Answer

The equation formed from the English sentence is \(y = x^2 + 2\). The graph of this equation is an upward-opening parabola that has its vertex at the point (0,2).

Step 1: Translate the English sentence into an equation

The sentence 'The \(y\) -value is two more than the square of the \(x\) -value' can be expressed as the equation \(y = x^2 + 2\). The square of the \(x\) -value is denoted by \(x^2\), and 'two more than' entails adding 2 to \(x^2\). Thus, the mathematical representation of the statement is \(y = x^2 + 2\).

Step 2: Graph the equation

To graph the equation \(y = x^2 + 2\), create a table of values for \(x\) and \(y\). Choose values for \(x\), substitute these values into the equation to find corresponding \(y\) values, and plot these (x, y) points. Connect the plotted points to form the graph. This equation represents a parabola that opens upwards and its vertex is at the point (0,2).

Key Concepts

Translating Verbal Expressions into Equations

Turning words into algebraic expressions is like translating one language into another. Start by identifying variables, which are often represented by letters like 'x' and 'y'. In our example, the 'x-value' and 'y-value' are the variables. The phrase 'two more than' implies addition. Therefore, when you read 'the y-value is two more than the square of the x-value,' you simply take the 'square of the x-value', written mathematically as \(x^2\), and add 2 to it to translate this into the equation \(y = x^2 + 2\).

This process involves recognizing keywords and understanding their corresponding mathematical operations. To get a better hold of this skill:

• Identify the operations: Addition (more than, added to), subtraction (less than, minus), multiplication (times, product of), and division (divided by, over).
• Understand the order of these operations.
• Match the operations with the correct algebraic symbols (+, -, *, /).

Practice makes perfect, so keep converting sentences to equations until it feels natural.

Quadratic Functions

Quadratic functions are like the sprinters of the algebra world – they always follow a particular path. These functions are represented by an equation of the form \(y = ax^2 + bx + c\), where 'a', 'b', and 'c' are constants, and 'a' is not equal to zero. The equation we're looking at, \(y = x^2 + 2\), is a simple form where 'a' is 1, 'b' is 0, and 'c' is 2.

These functions have unique features:

• The graph is a curve called a parabola.
• It has a highest or lowest point known as the vertex.
• The parabola can open upwards (if 'a' is positive) or downwards (if 'a' is negative).

In the classroom or for homework, you'll often be asked to find the vertex, the axis of symmetry (the line that splits the parabola into two mirror images), and the direction the parabola opens. Understanding these parts helps you to sketch the graph accurately.

Parabolas

Imagine a symmetrical arch, like the one a fountain makes with water – that's the shape of a parabola. Formally, it's the graph of a quadratic function, and it has distinctive features, such as a vertex and an axis of symmetry. As seen in our problem, the equation \(y = x^2 + 2\) graphs a parabola that opens upward. Its vertex is the point (0, 2), which is the lowest point since the parabola opens up.

Here’s what you need to know:

• The vertex can be found by evaluating the equation at the point where \(x\) makes the expression \(ax^2 + bx\) equal to zero.
• The axis of symmetry in this case is the vertical line \(x = 0\), since it's the line that runs through the vertex vertically.
• If you were to fold the graph along the axis of symmetry, both sides would match up perfectly.

Remembering these characteristics will help you to graph any quadratic function and understand its motion.