Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(y=x \sqrt{x+2}\)

Step-by-Step Solution

Answer

The x-intercept is 0 and the y-intercept is also 0.

Step 1: Find the x-intercepts

To find the x-intercepts, set the equation \(y=x \sqrt{x+2}\) equal to zero and solve for x. This gives the equation \[0 = x \sqrt{x+2}\]. The solutions of this equation are the x-intercepts.

Step 2: Solve the x-intercept equation

The equation \(0 = x \sqrt{x+2}\) can be solved by first expanding the equation into \(0 = x^2\sqrt{x}\) + \(2x\). This quadratic equation can be factored into x(x + 2) = 0. The solutions or roots are x = 0 and x = -2. Here, since we have \(\sqrt{x+2}\) in the original equation, we discard the solution \(x=-2\), as it will not produce a real result under the square root. So the x-intercept is x = 0.

Step 3: Find the y-intercepts

To find the y-intercepts, set x=0 in the original equation \(y=x \sqrt{x+2}\), we have \(y=0 \sqrt{2}\), which gives y = 0.

Key Concepts

Quadratic Equation

A quadratic equation is one of the foundational concepts in algebra and involves a polynomial equation of degree two. The general form of a quadratic equation is \[ ax^2 + bx + c = 0 \]where,

• \( a \), \( b \), and \( c \) are coefficients.
• \( a \) is not equal to zero, because if it were, the equation would not be quadratic but linear.
• The highest exponent of the variable \( x \) is two.

Quadratic equations can be solved by:

• Factoring,
• Completing the square,
• Using the quadratic formula,
• Graphing the equation.

Each method allows you to find the roots or solutions of the equation, which are the values of \( x \) that make the equation equal to zero. In our exercise, we used factoring to find the solutions for the x-intercepts.

Graphing

Graphing is a visual method to understand the behavior of equations. For a quadratic equation, the graph will be a parabola, which is a U-shaped curve. Depending on the sign of the leading coefficient \( a \), the parabola will either open upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)).Graphing involves:

• Identifying the vertex, which is the highest or lowest point of the parabola,
• Finding the axis of symmetry, which is a vertical line that splits the parabola into two mirror images,
• Locating the x-intercepts and y-intercepts. The x-intercepts are found where the graph crosses the x-axis, and the y-intercept is where it crosses the y-axis.

Understanding the graph of a quadratic equation helps you to quickly identify its intercepts and symmetry, giving insights into the solutions and the nature of the function.

Real Numbers

Real numbers include all the numbers that can be found on the number line. This includes both rational and irrational numbers.

• Rational numbers are numbers that can be expressed as the quotient or fraction of two integers. These include integers, fractions, and repeating or terminating decimals.
• Irrational numbers cannot be expressed as a simple fraction. These numbers have decimal expansions that neither terminate nor become periodic. Examples include \( \pi \) and \( \sqrt{2} \).

In our exercise, when identifying x-intercepts, it was necessary to consider whether the solutions resulted in real numbers. We discarded solutions that did not satisfy the condition of producing real results under the square root, ensuring our intercepts were indeed real numbers.

Square Root Function

A square root function is a function that involves the square root of a variable. It is generally represented as \( y = \sqrt{x} \).Key points about square root functions include:

• The domain includes only non-negative numbers since the square root of a negative number is not a real number.
• The range will consist of non-negative numbers as well, as square roots are non-negative.
• The graph of a square root function is a curve that starts at the origin and extends infinitely to the right, towards positive x-values.

In the given exercise, the term \( \sqrt{x+2} \) involves a square root function, which places a constraint on the values of \( x \). Specifically, the expression under the square root, \( x+2 \), must be greater than or equal to zero, ensuring the result is a real number. This affects how intercepts are calculated and what solutions are viable, as seen in discarding \( x=-2 \).