Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 2: Problem 31
Find the \(x\) - and \(y\) -intercepts of the graph of the equation. \(y=x \sqrt{x+2}\)
Short Answer
Expert verified
The x-intercept is 0 and the y-intercept is also 0.
Step by step solution
01
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.
02
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.
03
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.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
- Factoring,
- Completing the square,
- Using the quadratic formula,
- Graphing the equation.
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.
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 \( \sqrt{2} \).
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.
