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11.1: Graphs of Square Root Functions

Difficulty Level: At Grade Created by: CK-12
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You have used squared roots many times in this text: to simplify, to evaluate, and to solve. This lesson will focus on the graph of the square root function.

The square root function is defined by \begin{align*}f(x)=\sqrt{x-h}+k\end{align*}, where \begin{align*}x-h\ge0\end{align*} and \begin{align*}(h,k)\end{align*} represents the origin of the curve.

The graph of the parent function \begin{align*}f(x)=\sqrt{x}\end{align*} is shown below. The function is not defined for negative values of \begin{align*}x\end{align*}; you cannot take the square root of a negative number and get a real value.

By shifting the square root function around the coordinate plane, you will change the origin of the curve.

Example: Graph \begin{align*}f(x)=\sqrt{x}+4\end{align*} and compare it to the parent function.

Solution: This graph has been shifted vertically upward four units from the parent function \begin{align*}f(x)=\sqrt{x}\end{align*}. The graph is shown below.

Graphing Square Root Functions Using a Calculator

Graphing square root functions is similar to graphing linear, quadratic, or exponential functions. Use the following steps:

These figures should be side by side. Due to the captions, they have moved in a vertical alignment.

Practice Set

Sample explanations for some of the practice exercises below are available by viewing the following video. Note that there is not always a match between the number of the practice exercise in the video and the number of the practice exercise listed in the following exercise set.  However, the practice exercise is the same in both.

CK-12 Basic Algebra: Graphs of Square Root Functions (15:01)

  1. In the definition of a square root function, why must \begin{align*}(x-h) \ge 0\end{align*}?
  2. What is the domain and range of the parent function \begin{align*}f(x)=\sqrt{x}\end{align*}?

Identify the ordered pair of the origin of each square root function.

  1. \begin{align*}f(x)=\sqrt{x-2}\end{align*}
  2. \begin{align*}g(x)=\sqrt{x+4}+6\end{align*}
  3. \begin{align*}h(x)=\sqrt{x-1}-1\end{align*}
  4. \begin{align*}y=\sqrt{x}+3\end{align*}
  5. \begin{align*}f(x)=\sqrt{2x}+4\end{align*}

Graph the following functions on the same coordinate axes.

  1. \begin{align*}y=\sqrt{x}, \ y=2.5 \sqrt{x}\end{align*}, and \begin{align*}y=-2.5 \sqrt{x}\end{align*}
  2. \begin{align*}y=\sqrt{x}, \ y = 0.3 \sqrt{x}\end{align*}, and \begin{align*}y=0.6 \sqrt{x}\end{align*}
  3. \begin{align*}y=\sqrt{x}, \ y=\sqrt{x-5}\end{align*}, and \begin{align*}y=\sqrt{x+5}\end{align*}
  4. \begin{align*}y =\sqrt{x}, \ y = \sqrt{x} + 8\end{align*}, and \begin{align*}y=\sqrt{x}-8\end{align*}

In 12-20, graph the function.

  1. \begin{align*}y = \sqrt{2x-1}\end{align*}
  2. \begin{align*}y = \sqrt{4x+4}\end{align*}
  3. \begin{align*}y = \sqrt{5-x}\end{align*}
  4. \begin{align*}y = 2\sqrt{x}+5\end{align*}
  5. \begin{align*}y = 3-\sqrt{x}\end{align*}
  6. \begin{align*}y = 4 + 2\sqrt{x}\end{align*}
  7. \begin{align*}y = 2\sqrt{2x+3}+1\end{align*}
  8. \begin{align*}y = 4 + 2\sqrt{2-x}\end{align*}
  9. \begin{align*}y = \sqrt{x+1}-\sqrt{4x-5}\end{align*}
  10. The length between any two consecutive bases of a baseball diamond is 90 feet. How much shorter is it for the catcher to walk along the diagonal from home plate to second base than the runner running from second to home?
  11. The units of acceleration of gravity are given in feet per second squared. It is \begin{align*}g=32 \ ft/s^2\end{align*} at sea level. Graph the period of a pendulum with respect to its length in feet. For what length in feet will the period of a pendulum be two seconds?
  12. The acceleration of gravity on the Moon is \begin{align*}1.6 \ m/s^2\end{align*}. Graph the period of a pendulum on the Moon with respect to its length in meters. For what length, in meters, will the period of a pendulum be 10 seconds?
  13. The acceleration of gravity on Mars is \begin{align*}3.69 \ m/s^2\end{align*}. Graph the period of a pendulum on Mars with respect to its length in meters. For what length, in meters, will the period of a pendulum be three seconds?
  14. The acceleration of gravity on the Earth depends on the latitude and altitude of a place. The value of \begin{align*}g\end{align*} is slightly smaller for places closer to the Equator than places closer to the Poles, and the value of \begin{align*}g\end{align*} is slightly smaller for places at higher altitudes that it is for places at lower altitudes. In Helsinki, the value of \begin{align*}g=9.819 \ m/s^2\end{align*}, in Los Angeles the value of \begin{align*}g=9.796 \ m/s^2\end{align*}, and in Mexico City the value of \begin{align*}g=9.779 \ m/s^2\end{align*}. Graph the period of a pendulum with respect to its length for all three cities on the same graph. Use the formula to find the length (in meters) of a pendulum with a period of 8 seconds for each of these cities.
  15. The aspect ratio of a wide-screen TV is 2.39:1. Graph the length of the diagonal of a screen as a function of the area of the screen. What is the diagonal of a screen with area \begin{align*}150 \ in^2\end{align*}?

Graph the following functions using a graphing calculator.

  1. \begin{align*}y=\sqrt{3x-2}\end{align*}
  2. \begin{align*}y=4+\sqrt{2-x}\end{align*}
  3. \begin{align*}y = \sqrt{x^2-9}\end{align*}
  4. \begin{align*}y = \sqrt{x} - \sqrt{x+2}\end{align*}

Mixed Review

  1. Solve \begin{align*}16=2x^2-3x+4\end{align*}.
  2. Write an equation for a line with slope of 0.2 containing the point (1, 10).
  3. Are these lines parallel, perpendicular, or neither: \begin{align*}x+5y=16\end{align*} and \begin{align*}y=5x-3\end{align*}?
  4. Which of the following vertices minimizes the expression \begin{align*}20x+32y\end{align*}?
    1. (50, 0)
    2. (0, 60)
    3. (15, 30)
  5. Is the following graph a function? Explain your reasoning.
  6. Between which two consecutive integers is \begin{align*}\sqrt{205}\end{align*}?

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