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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 \(-4\) and the y-intercept is \(2\).

Step by step solution

01

Find the x-intercept

Set \(y=0\) in the equation. We then solve for \(x\): \[0=\sqrt{x+4}\] Squaring both sides to erase the square root gives: \[0=x+4\] Solving for \(x\) gives: \[x=-4\] So the x-intercept is \(-4\).
02

Find the y-intercept

Set \(x=0\) in the equation. Then we solve for \(y\): \[y=\sqrt{0+4}\] This simplifies to: \[y=\sqrt{4}=2\] So the y-intercept is \(2\).

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Key Concepts

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

x-intercept
The x-intercept of a graph is where the curve intersects the x-axis. In simpler terms, it's the point where the graph crosses the x-axis
  • To find the x-intercept, set the output variable, usually represented by \(y\), equal to zero.
  • So, for the equation \(y=\sqrt{x+4}\), substitute \(y=0\) and solve for \(x\). This results in \(0=\sqrt{x+4}\).
  • By squaring both sides, the equation becomes \(0=x+4\).
  • Solving for \(x\) yields \(x = -4\).
This means the graph of the function intersects the x-axis at the point \((-4, 0)\). This concept is crucial for graphing and understanding how functions behave as they approach the x-axis.
y-intercept
The y-intercept is the point at which a graph intersects the y-axis. In other words, it's the value of the output variable when the input variable \(x\) is zero.
  • To find the y-intercept, set \(x=0\) and solve for \(y\).
  • In the equation \(y=\sqrt{x+4}\), substitute \(x=0\) to get \(y=\sqrt{0+4}\).
  • This simplifies to \(y=\sqrt{4}\), which equals \(2\).
Thus, the y-intercept is at the point \((0, 2)\). This value shows where the graph will cross the y-axis, helping to predict the graph's position relative to the origin.
square root function
A square root function is a type of algebraic function that includes the square root of a variable, usually expressed by \(f(x)=\sqrt{x}\).
  • The function's domain is limited to non-negative values for the expression inside the square root, as square roots of negative numbers are not real.
  • The range typically starts at zero and goes to positive infinity, determined by the outputs of the square root expression.
  • For \(y=\sqrt{x+4}\), the domain is \(x \geq -4\) because the expression inside the square root, \(x+4\), must be non-negative.
Square root functions often have a unique curve that starts at a certain point and rises gently, reflecting the root's behavior. Understanding how these functions work is essential for accurate graphing and analysis.
graphing equations
Graphing equations visually demonstrates the relationship between variables, a pivotal skill in mathematics.
  • Start by identifying intercepts; they provide anchor points for sketching the curve.
  • Consider the function's shape, such as linear, quadratic, or square root, to predict its path.
  • For \(y=\sqrt{x+4}\), the graph starts at \(x=-4\) and moves upwards to the right.
When you graph equations like square root functions, only plot the portions where inputs result in real numbers. Use your x- and y-intercepts to sketch an accurate representation, as they guide the graph's shape and position. This practice not only furthers comprehending mathematical concepts but also aids in solving solvers that require visual interpretation.

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Most popular questions from this chapter

The suggested retail price of a new hybrid car is \(p\) dollars. The dealership advertises a factory rebate of \(\$ 2000\) and a \(10 \%\) discount. (a) Write a function \(R\) in terms of \(p\) giving the cost of the hybrid car after receiving the rebate from the factory. (b) Write a function \(S\) in terms of \(p\) giving the cost of the hybrid car after receiving the dealership discount. (c) Form the composite functions \((R \circ S)(p)\) and \((S \circ R)(p)\) and interpret each. (d) Find \((R \circ S)(25,795)\) and \((S \circ R)(25,795) .\) Which yields the lower cost for the hybrid car? Explain.

Prove that if \(f\) and \(g\) are one-to-one functions, then \((f \circ g)^{-1}(x)=\left(g^{-1} \circ f^{-1}\right)(x)\) .

Use the given values of \(k\) and \(n\) to complete the table for the direct variation model \(y=k x^{n} .\) Plot the points in a rectangular coordinate system. $$\begin{array}{|l|l|l|l|l|l|}\hline x & 2 & 4 & 6 & 8 & 10 \\\\\hline y=k x^{n} & & & & & \\\\\hline\end{array}$$ $$k=\frac{1}{4}, n=3$$

Determine whether the function has an inverse function. If it does, then find the inverse function. $$f(x)=|x-2|, \quad x \leq 2$$

Determine whether the function has an inverse function. If it does, then find the inverse function. $$f(x)=\frac{5 x-3}{2 x+5}$$
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