Warm Up 1) Use the graph to Identify the following: a)Vertex: b) Zeros: c) Range: d)Domain: e) Line of symmetry: f)Minimum/Maximum: (-2,-1) -3 and -1 R: y≥-1 D: all real numbers x=-2 Minimum=-1

Solving quadratics by graphing

21.0 Students graph quadratic functions and know that their roots are the x-intercepts. Also covered: 23.0 California Standards Objective

Two Zeros Steps for solving Quadratic Equations by Graphing One ZeroNo zeros Step 1: Write the related function. Step 2: Graph the function. Step 3: Find the zeros of the related function Two Roots One RootNo real Roots

Finding Roots of Quadratic Polynomials Find the roots of x 2 + 4x + 3 Step 1: Write the related equation. x 2 + 4x + 3=0y= x 2 + 4x + 3 Step 2: Graph the function. (make a T table) xy Step 3: Find the zeros of the related function The roots are at -1 and -3.

What is/are the root(s)?

You try: Solve the equation by graphing the related function. x x + 10 = 0 Step 1: Write the related function. Step 2: Graph the function.(make a T table) y= x x + 10 x (x,y) (-4 ) 2 +6(-4)+10=2 (-3 ) 2 +6(-1 ) +10=1 (-2 ) 2 +6(0) +10=2 (-1 ) 2 +6(-1) +10=5 (0 ) 2 +6(0)+10=-10 (-4,2) (-3,1) (-2,2) (-1,5) (0,10) Step 3: Find the zeros of the related function none

No zeros. Solve the equation by graphing the related function. 2 x 2 +4x = -3 Step 1: Write the related function. Step 2: Graph the function. (make a T table) Step 3: Find the zeros of the related function 2x x =-3 Hint: Move all terms to one side. +3 2x 2 +4x + 3 = 0 xy

Solve the equation by graphing the related function. 2 x 2 – 18 = 0 Step 1: Write the related function. Step 2: Graph the function.(make a T table) y= 2 x 2 – 18 xy=2x 2 -18(x,y) (-2 ) 2 -18=-10 2(-1 ) 2 -18=-16 2(-0 ) 2 -18=-18 2(-1 ) 2 -18=-16 2(2 ) 2 -18=-10 (-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. x 2 – 8x – 16 = 2x 2 Step 1: Write the related function. x x-16=2x 2 Hint: Move all terms to one side. You want a positive leading coefficient -x 2 +8x+16 0= x 2 + 8x +16 Step 2: Graph the function. (make a T table) xy Step 3: Find the zeros of the related function The only zero appears to be -4.

The only zero appears to be 3. Solve the equation by graphing the related function. -12 x +18 = -2 x 2 Step 1: Write the related function. Step 2: Graph the function. (make a T table) Step 3: Find the zeros of the related function -12x + 18 x=-2x 2 Hint: Move all terms to one side. You want a positive leading coefficient +2x 2 2x 2 – 12x + 18 = 0 xy

1) x 2 -x+20 2) x 2 -12x+35 3) x 2 +x-2 4) 9x 2 -6x+2 5) x 2 -4x+4 6) A frog jumps straight up from the ground. The quadratic function f(t) = –16t t models the frog’s height above the ground after t seconds. About how long is the frog in the air? Hint: When the frog leaves the ground, its height is 0, and when the frog lands, its height is 0. So solve 0 = –16t t to find the times when the frog leaves the ground and lands. -4 and 5 5 and 7 1 and -2 none 2 0 and 0.75

Lesson Quiz Solve each equation by graphing the related function. 1.3x 2 – 12 = 0 2.x 2 + 2x = 8 3.3x – 5 = x 2 4.3x = 6x 5.A rocket is shot straight up from the ground. The quadratic function f(t) = –16t t models the rocket’s height above the ground after t seconds. How long does it take for the rocket to return to the ground? 2, –2 –4, 2 ø 1 6 s