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Warm Up 1) Use the graph to Identify the following: a)Vertex: (-2,-1) b) Zeros: -3 and -1 c) Range: R: y≥-1 d)Domain: D: all real numbers e) Line of symmetry: x=-2 f)Minimum/Maximum: Minimum=-1 Solving quadratics by graphing Objective California Standards 21.0 Students graph quadratic functions and know that their roots are the x-intercepts. Also covered: 23.0 Steps for solving Quadratic Equations by Graphing Step 1: Write the related function. Step 2: Graph the function. Step 3: Find the zeros of the related function Two Zeros One Zero No zeros Two Roots One Root No real Roots Finding Roots of Quadratic Polynomials Find the roots of x2 + 4x + 3 Step 1: Write the related equation. x2 + 4x + 3=0 y= x2 Step 2: Graph the function. (make a T table) Step 3: Find the zeros of the related function The roots are at -1 and -3. + 4x + 3 x -4 y 3 -3 0 -2 -1 -1 0 0 3 What is/are the root(s)? You try: Solve the equation by graphing the related function. x2 + 6x + 10 = 0 Step 1: Write the related function. y= x2 + 6x + 10 Step 2: Graph the function.(make a T table) x -4 -3 y=x2+6x+10 (-4 )2+6(-4)+10=2 (-3 )2+6(-1 ) +10=1 -2 (-2 )2+6(0) +10=2 -1 (-1 )2+6(-1) +10=5 0 (0 )2+6(0)+10=-10 (x,y) (-4,2) (-3,1) (-2,2) (-1,5) (0,10) Step 3: Find the zeros of the related function none Solve the equation by graphing the related function. 2x2 +4x = -3 Step 1: Write the related function. 2x2+ 4x =-3 +3 +3 Hint: Move all terms to one side. 2x2 +4x + 3 = 0 Step 2: Graph the function. (make a T table) Step 3: Find the zeros of the related function No zeros. x -3 y 9 -2 3 -0 1 1 3 2 9 Solve the equation by graphing the related function. 2x2 – 18 = 0 Step 1: Write the related function. y= 2x2 – 18 Step 2: Graph the function.(make a T table) x y=2x2-18 -2 2(-2 )2-18=-10 -1 2(-1 )2-18=-16 0 2(-0 )2-18=-18 1 2(-1 )2-18=-16 2 2(2 )2-18=-10 (x,y) (-2,-10) (-1,-16) (0,-18) (1,-16) (2,-10) Step 3: Find the zeros of the related function -3 and 3 You try: Solve the equation by graphing the related function. x2 – 8x – 16 = 2x2 Step 1: Write the related function. 2- 8x-16=2x2 x -x2+8x+16 -x2+8x+16 0= x2+ 8x +16 x -5 y 1 Hint: Move all terms to one side. You want a positive -4 leading 0 coefficient Step 2: Graph the function. (make a T table) Step 3: Find the zeros of the related function The only zero appears to be -4. -3 1 -2 4 -1 9 Solve the equation by graphing the related function. -12x +18 = -2x2 Step 1: Write the related function. -12x + +2x2 18x=-2x2 +2x2 x 1 Hint: Move all terms to one side. You want a positive leading 2 coefficient 2x2 – 12x + 18 = 0 Step 2: Graph the function. (make a T table) Step 3: Find the zeros of the related function The only zero appears to be 3. y 8 2 3 0 4 2 5 8 Using your graphing calculator: Find the roots of each quadratic polynomial -4 and 5 6) A frog jumps straight up from the ground. The 1) x2-x+20 5 and 7 quadratic function 2 2) x -12x+35 2 + 12t models f(t) = –16t 1 and -2 the frog’s height above 3) x2+x-2 none the ground after t 4) 9x2-6x+2 seconds. About how long 2 is the frog in the air? 2 5) x -4x+4 0 and 0.75 Hint: When the frog leaves the ground, its height is 0, and when the frog lands, its height is 0. So solve 0 = –16t2 + 12t to find the times when the frog leaves the ground and lands. Lesson Quiz Solve each equation by graphing the related function. 1. 3x2 – 12 = 0 2, –2 2. x2 + 2x = 8 –4, 2 3. 3x – 5 = x2 ø 4. 3x2 + 3 = 6x 1 5. A rocket is shot straight up from the ground. The quadratic function f(t) = –16t2 + 96t models the rocket’s height above the ground after t seconds. How long does it take for the rocket to return to the ground? 6 s