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Chapter 1: Problem 25

Find the \(x\) - and \(y\) -intercepts of the graph of the equation. $$ y=\sqrt{x+4} $$

Short Answer

Expert verified
The x-intercept is (-4,0) and the y-intercept is (0,2).

Step by step solution

01

Find the x-intercept

To find the x-intercept, set \(y=0\) in the equation \(y=\sqrt{x+4}\). Solve this equation for \(x\). That gives us \(0=\sqrt{x+4}\), thus \(x=-4\). Therefore, the x-intercept of the function is (-4,0).
02

Find the y-intercept

To find the y-intercept, set \(x=0\) in the equation \(y=\sqrt{x+4}\). Solve this equation for \(y\), which results in \(y=\sqrt{0+4}\) implies \(y=2\). Therefore, the y-intercept of the function is (0,2).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding x-intercepts
When working with graphs, the **x-intercept** is a crucial point where the graph crosses the x-axis. This indicates the value of \( x \) when \( y = 0 \). To find the x-intercept for the equation \( y = \sqrt{x+4} \), we start by setting \( y \) equal to zero. This simplifies to \( 0 = \sqrt{x+4} \).
  • This implies \( x + 4 = 0 \) because the square root of a number results in zero only if that number itself is zero.
  • Solving this equation gives \( x = -4 \).
Thus, the x-intercept is the point \( (-4, 0) \). This means the graph crosses the x-axis at \( x = -4 \). This process can be used for any function to find where it intersects the x-axis. Always remember, for the x-intercept, \( y \) is zero.
The significance of y-intercepts
While x-intercepts show where a graph crosses the x-axis, a **y-intercept** represents where the graph crosses the y-axis. This point reflects the value of \( y \) when \( x \) is zero. Let's find the y-intercept for the equation \( y = \sqrt{x+4} \), by setting \( x \) to zero. This gives us:\[ y = \sqrt{0 + 4} \]
  • Solving for \( y \), we find \( y = \sqrt{4} = 2 \).
  • Thus, the y-intercept is at the point \( (0, 2) \).
In essence, the graph intersects the y-axis at \( y = 2 \). The y-intercept gives insight into the starting value of the function when no other variable is affecting it.
Exploring radical equations
**Radical equations** involve variables within a radical, usually a square root. In this context, the equation \( y = \sqrt{x+4} \) falls under this category. Radical equations can often have certain limitations and characteristics:
  • The value under a square root sign has to be non-negative, meaning \( x+4 \geq 0 \), so \( x \geq -4 \). This prevents having an undefined solution in real numbers.
  • It's beneficial to analyze these limitations when graphing or solving, as they affect the domain and range of the function.
  • Radical equations often appear in forms where finding x- and y-intercepts requires solving by squaring both sides strategically, as shown in previous sections.
Understanding how these types of equations behave can be very helpful in identifying the intercepts and limits they impose on a graph. Always consider checking your results, as extraneous solutions can appear when working with radicals.

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Most popular questions from this chapter

Use the functions given by \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$ (g \circ f)^{-1} $$

Use the functions given by \(f(x)=x+4\) and \(g(x)=2 x-5\) to find the specified function. $$ g^{-1} \circ f^{-1} $$

The simple interest on an investment is directly proportional to the amount of the investment. By investing \(\$ 3250\) in a certain bond issue, you obtained an interest payment of \(\$ 113.75\) after 1 year. Find a mathematical model that gives the interest \(I\) for this bond issue after 1 year in terms of the amount invested \(P\)

Determine whether the function has an inverse function. If it does, find the inverse function. $$ f(x)=\left\\{\begin{array}{ll} -x, & x \leq 0 \\ x^{2}-3 x, & x>0 \end{array}\right. $$

The total sales (in billions of dollars) for CocaCola Enterprises from 2000 through 2007 are listed below. (Source: Coca-Cola Enterprises, Inc.) $$\begin{array}{llll} 2000 & 14.750 & 2004 & 18.185 \\ 2001 & 15.700 & 2005 & 18.706 \\ 2002 & 16.899 & 2006 & 19.804 \\ 2003 & 17.330 & 2007 & 20.936 \end{array}$$ (a) Sketch a scatter plot of the data. Let \(y\) represent the total revenue (in billions of dollars) and let \(t=0\) represent 2000 . (b) Use a straightedge to sketch the best-fitting line through the points and find an equation of the line. (c) Use the regression feature of a graphing utility to find the least squares regression line that fits the data. (d) Compare the linear model you found in part (b) with the linear model given by the graphing utility in part (c). (e) Use the models from parts (b) and (c) to estimate the sales of Coca-Cola Enterprises in 2008 . (f) Use your school's library, the Internet, or some other reference source to analyze the accuracy of the estimate in part (e).
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