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Vertex Form

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Forms of quadratics Factored form a(x-r 1 )(x-r 2 ) Standard Form ax 2 +bx+c Vertex Form a(x-h) 2 +k

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Each form gives you different information! Factored form a(x-r 1 )(x-r 2 ) – Tells you direction of opening – Tells you location of x-intercepts (roots) Standard Form ax 2 +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)

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Direction of opening x 2 opens up

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Direction of opening ax 2 stretches x vertically by a – Here a is 1.5

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Direction of opening ax 2 stretches x vertically by a – Here a is 0.5 – Stretching by a fraction is a squish

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Direction of opening ax 2 stretches x vertically by a – Here a is -0.5 – Stretching by a negative causes a flip

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Direction of opening a is the number in front of the x 2 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-r 1 )(x-r 2 ) – ax 2 +bx+c – a(x-h) 2 +k

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Factored form a(x-r 1 )(x-r 2 ) a is the direction of opening r 1 and r 2 are the x-intercepts – Or roots, or zeros Example: -2(x-2)(x+0.5) – a is negative, opens down. – r 1 is 2, crosses the x-axis at 2. – r 2 is -0.5, crosses the x-axis at -0.5

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Factored form a(x-r 1 )(x-r 2 ) a is the direction of opening r 1 and r 2 are the x-intercepts – Or roots, or zeros Example: -2(x-2)(x+0.5) – a is negative, opens down. – r 1 is 2, crosses the x-axis at 2. – r 2 is -0.5, crosses the x-axis at -0.5

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Standard form ax 2 +bx+c a is the direction of opening c is the y-intercept – ƒ(0)=a0 2 +b0+c=c Example: -2x 2 +3x+2 – Opens down – Crosses through the point (0,2)

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Standard form ax 2 +bx+c a is the direction of opening c is the y-intercept – ƒ(0)=a0 2 +b0+c=c Example: -2x 2 +3x+2 – Opens down – Crosses through the point (0,2)

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Vertex form Start with f(x)=x 2

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Vertex form Stretch/Flip if you want – aƒ(x)=ax 2

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Vertex form Shift right by h – aƒ(x-h)=a(x-h) 2 h

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Vertex form Shift up by k – aƒ(x-h)+k=a(x-h) 2 +k h k

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Vertex form Define a new function – g(x)=a(x-h) 2 +k (h,k)

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Vertex form a(x-h) 2 +k a tells you direction of opening (h,k) is the vertex (h,k)

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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)

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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)

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Switching between forms Gives you a full picture Example: ƒ(x)=-2(x-2)(x+0.5) ƒ(x)=-2x 2 +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)

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Switching between forms Gives you a full picture Example: ƒ(x)=-2(x-2)(x+0.5) ƒ(x)=-2x 2 +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)

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Consider the function f(x) = -3x 2 +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.

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Consider the function f(x) = -3x 2 +2x-9. Which of the following are true? Standard form: ax 2 +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.

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The Vertex Formula Remember the Quadratic formula

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What does the QF say?

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The Vertex Formula

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Example

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Given the function R(x)=(2x+6)(x-12), find an equation for its axis of symmetry. A)x = - 9 B)x = 9 C)x = 2 D)x = 6 E)None of the above.

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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.

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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?

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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) (x 1,y 1 )=(0,1)

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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) (x 1,y 1 )=(0,1) But to be finished, I need to know a! Use: My formula is true for every x,y including x 1,y 1

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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) (x 1,y 1 )=(0,1) My formula is true for every x,y; not just x 1,y 1

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A quadratic function has vertex at (0,2) and passes through the point (1,3). Find an equation for this parabola. A)y = (x+2) 2 B)y = x 2 +3 C)y = x 2 +1 D)y = x 2 E)None of the above

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A quadratic function has vertex at (0,2) and passes through the point (1,3). Find an equation for this parabola. E

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