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Vertex Form Forms of quadratics • Factored form a(x-r1)(x-r2) • Standard Form ax2+bx+c • Vertex Form a(x-h)2+k Each form gives you different information! • Factored form a(x-r1)(x-r2) – Tells you direction of opening – Tells you location of x-intercepts (roots) • Standard Form ax2+bx+c – Tells you direction of opening – Tells you location of y-intercept • Vertex Form a(x-h)2+k – Tells you direction opening – Tells you the location of the vertex (max or min) Direction of opening • x2 opens up Direction of opening • ax2 stretches x vertically by a – Here a is 1.5 Direction of opening • ax2 stretches x vertically by a – Here a is 0.5 – Stretching by a fraction is a squish Direction of opening • ax2 stretches x vertically by a – Here a is -0.5 – Stretching by a negative causes a flip Direction of opening • a is the number in front of the x2 • The value a tells you what direction the parabola is opening in. – Positive a opens up – Negative a opens down • The a in all three forms is the same number – a(x-r1)(x-r2) – ax2+bx+c – a(x-h)2+k Factored form a(x-r1)(x-r2) • a is the direction of opening • r1 and r2 are the x-intercepts – Or roots, or zeros • Example: -2(x-2)(x+0.5) – a is negative, opens down. – r1 is 2, crosses the x-axis at 2. – r2 is -0.5, crosses the x-axis at -0.5 Factored form a(x-r1)(x-r2) • a is the direction of opening • r1 and r2 are the x-intercepts – Or roots, or zeros • Example: -2(x-2)(x+0.5) – a is negative, opens down. – r1 is 2, crosses the x-axis at 2. – r2 is -0.5, crosses the x-axis at -0.5 Standard form ax2+bx+c • a is the direction of opening • c is the y-intercept – ƒ(0)=a02+b0+c=c • Example: -2x2+3x+2 – Opens down – Crosses through the point (0,2) Standard form ax2+bx+c • a is the direction of opening • c is the y-intercept – ƒ(0)=a02+b0+c=c • Example: -2x2+3x+2 – Opens down – Crosses through the point (0,2) Vertex form • Start with f(x)=x2 Vertex form • Stretch/Flip if you want – aƒ(x)=ax2 Vertex form • Shift right by h – aƒ(x-h)=a(x-h)2 h Vertex form • Shift up by k – aƒ(x-h)+k=a(x-h)2+k k h Vertex form • Define a new function – g(x)=a(x-h)2+k (h,k) Vertex form a(x-h)2+k • a tells you direction of opening • (h,k) is the vertex (h,k) Vertex form a(x-h)2+k • a tells you direction of opening • (h,k) is the vertex • Example: -2(x-3/4)2+25/8 – Opens down – Has vertex at (3/4, 25/8) Vertex form a(x-h)2+k • a tells you direction of opening • (h,k) is the vertex • Example: -2(x-3/4)2+25/8 – Opens down – Has vertex at (3/4, 25/8) (3/4, 25/8) Switching between forms Gives you a full picture • Example: ƒ(x)=-2(x-2)(x+0.5) ƒ(x)=-2x2+3x+2 ƒ(x)=-2(x-3/4)2+25/8 are all the same function – Opens down – Crosses x axis at 2 and -0.5 – Crosses the y-axis at 2 – Has vertex at (3/4, 25/8) Switching between forms Gives you a full picture • Example: ƒ(x)=-2(x-2)(x+0.5) ƒ(x)=-2x2+3x+2 ƒ(x)=-2(x-3/4)2+25/8 are all the same function – Opens down – Crosses x axis at 2 and -0.5 – Crosses the y-axis at 2 – Has vertex at (3/4, 25/8) Consider the function f(x) = -3x2+2x-9. Which of the following are true? A) The graph of f(x) has a negative y-intercept B) f(x) has 2 real zeros. C) The graph of f(x) attains a maximum value D) Both (A) and (B) are true E) Both (A) and (C) are true. Consider the function f(x) = -3x2+2x-9. Which of the following are true? Standard form: ax2+bx+c. a is negative: opens down. ƒ(x) attains a maximum value. (C) is true. c is my y-intercept. c is negative. My y-intercept is negative. (A) is true. E) Both (A) and (C) are true. The Vertex Formula • Remember the Quadratic formula when ax + bx + c = 0 2 -b b - 4ac x= ± 2a 2a 2 What does the QF say? + - b 2 - 4ac 2a Is the distance you have to move b 2 - 4ac 2a b 2 - 4ac 2a from the center left and right to get to the roots x= -b is the line of symmetry for the curve 2a The Vertex Formula when f (x) = ax 2 + bx + c And you want to rewrite f (x) as f (x) = a(x - h)2 + k -b h= 2a and k = f (h) Example when f (x) = -2x 2 + 3x + 2 And you want to rewrite f (x) as f (x) = a(x - h)2 + k -3 3 h= = 2(-2) 4 and k = f ( ) = -2 ( 3 4 3 4 ) 2 + 3( 34 ) + 2 = -2 ( 169 ) + 94 + 2 = -98 + 188 + 168 = 258 So f (x) = -2 ( x - 3 4 ) 2 + 258 Given the function R(x)=(2x+6)(x-12), find an equation for its axis of symmetry. A) B) C) D) E) x=-9 x=9 x=2 x=6 None of the above. Given the function R(x)=(2x+6)(x-12), find an equation for its axis (line) of symmetry. • The roots are x=-3 and x=12. • The axis of symmetry is halfway between the roots. • (12-3)/2=4.5, the number halfway between -3 and 12. • x=4.5 is the axis of symmetry • E) None of the above. How to find an equation from vertex and point • A parabola passes has its vertex at (1,3) and passes through the point (0,1). What is the equation of this parabola? How to find an equation from vertex and point • A parabola passes has its vertex at (1,3) and passes through the point (0,1). What is the equation of this parabola? • (h,k)=(1,3) • (x1,y1)=(0,1) How to find an equation from vertex and point 2 • A parabola y = a(x - h) + k passes has its vertex at (1,3) and passes through the point (0,1). What is the equation of this parabola? • (h,k)=(1,3) • (x1,y1)=(0,1) y = a(x -1) + 3 2 But to be finished, I need to know a! Use: My formula is true for every x,y including x1,y1 How to find an equation from vertex and point 2 y = a(x - h) + k • A parabola passes has its vertex at (1,3) and passes through the point (0,1). What is the equation of this parabola? • (h,k)=(1,3) • (x1,y1)=(0,1) My formula is true for every x,y; not just x1,y1 y = a(x -1) + 3 2 y1 = a(x1 -1) + 3 2 1 = a(0 -1) + 3 -2 = a(1) a = -2 2 y = -2(x -1) + 3 2 A quadratic function has vertex at (0,2) and passes through the point (1,3). Find an equation for this parabola. A) B) C) D) E) y = (x+2)2 y = x2+3 y = x2+1 y = x2 None of the above A quadratic function has vertex at (0,2) and passes through the point (1,3). Find an equation for this parabola. Generic Formula: y = a(x - h) + k 2 Plug in vertex: y = a(x - 0) + 2 2 y = ax 2 + 2 Find a: 3 = a12 + 2 1= a Plug in a: y = 1x 2 + 2 y= x +2 2 E