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ALGEBRA 1 “Graphing Square Root Functions” What is a “square root function”? Square Root Function: function, f(x), in which the independent variable, x, is inside a radicand Example: y = x (the simplest square root function) y = x+3 y = 2x - 8 How do you find all of the solutions to a square root function? Since a radicand can’t be negative (you can’t square a negative value), the domain (x values) is limited to those values of x that make the radicand greater than or equal to zero. Therefore, you can create a function table to graph the solutions with x being values that make the expression inside the radicand greater than or equal to zero. To find the minimum value of the range, plug in the minimum value of x that will make the expression in the radical equal zero. Examples: Subtract 3 or add 8 to both sides Make a function table. ALGEBRA 1 Recall that the square root of a negative number is not a real number. The domain (x-values) of a square-root function is restricted to numbers that make the value under the radical sign greater than or equal to 0. Key Rules: *The domain and range are the set of nonnegative real numbers. *The x-intercept and the y-intercept are at (0, 0). *The function is neither even nor odd. *It is increasing on the interval (0, ). *It has a minimum value of 0 at x = 0. ALGEBRA 1 Graphing Square Root Functions LESSON 10-5 Additional Examples I Find the domain of each function. a. y = x+5 x+5> –0 Make the radicand > – 0. x> – –5 The domain is the set of all numbers greater than or equal to –5. b. y = 6 4x – 12 4x – 12 > –0 Make the radicand > – 0. 4x > – 12 x> – 3 The domain is the set of all numbers greater than or equal to 3. ALGEBRA 1 Find the domain of the square-root function. x– 4 ≥ 0 x +3 ≥ –3 +4 +4 x ≥ 4 The domain is the set of all real numbers greater than or equal to 4. x 0 –3 ≥ –3 The domain is the set of all real numbers greater than or equal to –3. ALGEBRA 1 Find the domain of the square-root function. 2x – 1 +1 ≥ 0 +1 2x ≥ 1 The domain is the set of all real numbers greater than or equal to . 3x – 5 +5 3x ≥ 0 +5 ≥ 5 The domain is the set of all real numbers greater than or equal to . ALGEBRA 1 “Graphing Square Root Functions” I How do you graph a square root function? To graph a square root function: 1. Make a function table with at least four x values. Make one of the x values the minimum value of x that makes the expression inside the radicand equal zero. 2. Plot the x and y value on the coordinate grid. 3. Connect the points to form a curve (should look like half of a parabola) Examples: Step 1. Function Table Step 2. Plot the points. Step 3. Connect the points to form a half parabola shape Notice there are no x-values to the left of 0 because the domain is x ≥ 0. ALGEBRA 1 y a xh k f ( x) x Add a positive number to x. f ( x) x h Shift left h. Add a negative number to x. f ( x) x h Shift right h. Add a positive Add a negative number to number the toAdd the radical. a positive number to the radical. radical. f ( x) x k f ( x) x k Up k. Down k. ALGEBRA 1 If a square-root function is given in one of these forms, you can graph the parent function and translate it vertically or horizontally. ALGEBRA 1 Graph . Since this function is in the form f(x) =, you can graph it as a horizontal translation of the graph of f(x) = Graph f(x) = and then shift the graph 3 units to the right. ALGEBRA 1 Graph each square root function. Since this function is in the form f(x) = , you can graph it as a vertical translation of the graph of f(x) = Graph f(x) = and then shift the graph 2 units up. ALGEBRA 1 Graph each square-root function. A. B. ALGEBRA 1 Graphing Square Root Functions LESSON 10-5 Additional Examples Graph y = y= x + 4 by translating the graph of x. For the graph y = the graph of y = x + 4, x is shifted 4 units up. ALGEBRA 1 Graphing Square Root Functions LESSON 10-5 Additional Examples Graph ƒ(x) = y= x + 3 by translating the graph of x. For the graph ƒ(x) = the graph of y = x + 3, x is shifted to the left 3 units. ALGEBRA 1 I Graph Step 1 Choose x-values greater than or equal to 0 and generate ordered pairs. . Step 2 Plot the points. Then connect them with a smooth curve. x 0 4 1 7 4 10 6 11.35 ALGEBRA 1 Graph . Step 1 Choose x-values greater than or equal to 0 and generate ordered pairs. x 0 1 4 6 Step 2 Plot the points. Then connect them with a smooth curve. 3 5 7 7.89 ALGEBRA 1 Graphing Square Root Functions LESSON 10-5 1. Find the domain of the function ƒ(x) = A. Graph y = 3 B. Graph y = 2x – 4. x>2 x. x – 3. 4. Describe how to translate the graph of y = x to obtain the graph of the function y = x – 15. Shift the graph to the right 15 units. ALGEBRA 1 Graphing Square-root Functions Graphing y a x k Domain: all nonnegative numbers Range: all numbers greater than or equal to k 1. y 2 x 1 Domain: x 0 Range: y 1 x y 0 y 2 0 1 1 1 y 2 1 1 3 4 y 2 4 1 5 9 y 2 9 1 7 ALGEBRA 1 Graphing Square-root Functions Graphing y a x k 2. y 2 x 1 Domain: x 0 Range: y 1 x y 0 y 2 0 1 1 1 y 2 1 1 1 4 y 2 4 1 3 9 y 2 9 1 5 ALGEBRA 1 Graphing Square-root Functions Graphing y a x k 3. y 2 x 2 Domain: x 0 Range: y 2 x y 0 y 2 0 2 2 1 y 2 120 4 y 2 4 22 9 y 2 9 24 ALGEBRA 1 Graphing Square-root Functions Graphing y x h Domain: all numbers greater than or equal to h Range: all nonnegative numbers 4. y x 1 Domain: x 1 Range: y 0 x y 1 y 11 0 2 y 2 1 1 5 y 51 2 10 y 10 1 3 ALGEBRA 1 Graphing Square-root Functions Graphing y x h 5. y x 1 Domain: x 1 Range: y 0 x y -1 y 1 1 0 0 y 0 1 1 3 y 31 2 8 y 8 1 3 ALGEBRA 1 Graphing Square-root Functions Graphing y x h 6. y x 4 Domain: x 4 Range: y 0 x y 4 y 44 0 5 y 5 4 1 8 y 84 2 13 y 13 4 3 ALGEBRA 1 Graphing Square-root Functions Compare the Graphs y x 0 1 4 9 x k yya xxhhkk y 0 1 2 3 y 3 x32 ALGEBRA 1 ALGEBRA 1