Algebra1 Square-Root Functions CONFIDENTIAL

Warm Up 1) Blake invested $42,000 at a rate of 5% compounded quarterly. Write a function to model this situation. Then find the value of Blake’s investment after 3 years. 2) Lead-209 has a half-life of about 3.25 hours. Find the amount of lead-209 left from a 230-mg sample after 1 day. Round your answer to the nearest hundredth. 1) A = 42,000(1.0125)4t ; $48751.69 2) 1.38g CONFIDENTIAL

Square-Root Functions A square-root function is a function whose rule contains a variable under a square-root sign. EXAMPLES y = √x y = 2x + 1 y = 3 – x - 6 2 NONEXAMPLES y = x2 y = 2 . x + 1 y = √ 3 x CONFIDENTIAL

Square-Root Functions A) Find the speed of an object in free fall after it has fallen 4 feet. y = 8 √x = 8 √4 = 8 (2) = 16 Write the speed function. Substitute 4 for x. Simplify. After an object has fallen 4 feet, its speed is 16 ft/s. CONFIDENTIAL

After an object has fallen 50 feet, its speed is about 56.6 ft/s. B) Find the speed of an object in free fall after it has fallen 50 feet. Round your answer to the nearest tenth. y = 8 √x = 8 √50 ≈ 56.6 Write the speed function. Substitute 50 for x. Use a calculator. After an object has fallen 50 feet, its speed is about 56.6 ft/s. CONFIDENTIAL

Now you try! 1a) Find the speed of an object in free fall after it has fallen 25 feet. 1b) Find the speed of an object in free fall after it has fallen 15 feet. Round your answer to the nearest hundredth. 1a) 40 ft/s 1b) 30.98 ft/s CONFIDENTIAL

make the value under the radical sign greater than or equal to 0. 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. CONFIDENTIAL

Square-Root Functions Find the domain of each square-root function. A) y = x + 4 - 3 x + 4 ≥ 0 The expression under the radical sign must be greater than or equal to 0. - 4 - 4 Solve the inequality. Subtract 4 from both sides. x ≥ -4 The domain is the set of all real numbers greater than or equal to -4. CONFIDENTIAL

The domain is the set of all real numbers greater than or equal to 2. B) y = 3 ( x – 2) 3 ( x – 2) ≥ 0 The expression under the radical sign must be greater than or equal to 0. Solve the inequality. Distribute 3 on the left side. 3x – 6 ≥ 0 + 6 + 4 Add 6 to both sides. 3x ≥ 6 x ≥ 2 Divide both sides by 3. The domain is the set of all real numbers greater than or equal to 2. CONFIDENTIAL

Find the domain of each square-root function. Now you try! Find the domain of each square-root function. 2a) y = 2 x - 1 2b) y = 3x - 5 2a) x ≥ 1/2 2b) x ≥ 5/3 CONFIDENTIAL

The parent function for square-root functions, f (x) = √x , is graphed at right. Notice there are no x-values to the left of 0 because the domain is x ≥ 0. CONFIDENTIAL

Translations of the Graph of f (x) = √x If a square-root function is given in one of these forms, you can graph the parent function f(x) = √x and translate it vertically or horizontally. CONFIDENTIAL

Graphing Square-Root Functions A) Graph f (x) = x – 4 Since this function is in the form f (x) = x - a , you can graph it as a horizontal translation of the graph of f (x) = √x . Graph f (x) = √x and then shift the graph 4 units to the right. CONFIDENTIAL

Step1: Find the domain of the function. B) Graph f (x) = 2x + 3 This is not a horizontal or vertical translation of the graph of f (x) = √ x . Step1: Find the domain of the function. The expression under the radical sign must be greater than or equal to 0. 2x ≥ 0 Solve the inequality by dividing both sides by 2. x ≥ 0 The domain is the set of all real numbers greater than or equal to 0. CONFIDENTIAL

Step3: Plot the points. Then connect them with a smooth curve. Step2: Choose x-values greater than or equal to 0 and generate ordered pairs. Step3: Plot the points. Then connect them with a smooth curve. CONFIDENTIAL

Graph each square-root function. Now you try! Graph each square-root function. 3a) f (x) = √x + 2 3b) f (x) = 2√x + 3 CONFIDENTIAL

Assessment 1) Explain why y = x + √3 is not a square-root function. 2) In a right triangle, c = a2 + b2 , where c is the length of the hypotenuse (the longest side) and a and b are the lengths of the other two sides, called the legs. What is the length of the hypotenuse of a right triangle if its legs measure 14 cm and 8 cm? Round your answer to the nearest hundredth. 1) There is no variable under the square root sign. 2) 16.12 cm CONFIDENTIAL

Find the domain of each square-root function. 3) y = x + 6 4) y = 4 - 3 - x 5) y = 2x - 5 6) y = x + 2 7) y = 3x + 9 8) y = x + x - 5 3) x ≥ -6 4) x ≤ 3 5) x ≥ 0 6) x ≥ -2 7) x ≥ -3 8) x ≥ 5 CONFIDENTIAL

Graph each square-root function. 9) y = x - 1 10) y = 2x CONFIDENTIAL

Square-Root Functions Let’s review Square-Root Functions A square-root function is a function whose rule contains a variable under a square-root sign. EXAMPLES y = √x y = 2x + 1 y = 3 – x - 6 2 NONEXAMPLES y = x2 y = 2 . x + 1 y = √ 3 x CONFIDENTIAL

Square-Root Functions A) Find the speed of an object in free fall after it has fallen 4 feet. y = 8 √x = 8 √4 = 8 (2) = 16 Write the speed function. Substitute 4 for x. Simplify. After an object has fallen 4 feet, its speed is 16 ft/s. CONFIDENTIAL

Square-Root Functions Find the domain of each square-root function. A) y = x + 4 - 3 x + 4 ≥ 0 The expression under the radical sign must be greater than or equal to 0. - 4 - 4 Solve the inequality. Subtract 4 from both sides. x ≥ -4 The domain is the set of all real numbers greater than or equal to -4. CONFIDENTIAL

The parent function for square-root functions, f (x) = √x , is graphed at right. Notice there are no x-values to the left of 0 because the domain is x ≥ 0. CONFIDENTIAL

Translations of the Graph of f (x) = √x If a square-root function is given in one of these forms, you can graph the parent function f(x) = √x and translate it vertically or horizontally. CONFIDENTIAL

Step1: Find the domain of the function. B) Graph f (x) = 2x + 3 This is not a horizontal or vertical translation of the graph of f (x) = √ x . Step1: Find the domain of the function. The expression under the radical sign must be greater than or equal to 0. 2x ≥ 0 Solve the inequality by dividing both sides by 2. x ≥ 0 The domain is the set of all real numbers greater than or equal to 0. CONFIDENTIAL

Step3: Plot the points. Then connect them with a smooth curve. Step2: Choose x-values greater than or equal to 0 and generate ordered pairs. Step3: Plot the points. Then connect them with a smooth curve. CONFIDENTIAL

You did a great job today! CONFIDENTIAL