Vertex and Intercept Form of Quadratic Function Standard: MM2A3c Students will Investigate and explain characteristics of quadratic functions, including domain, range, vertex, axis of symmetry, zeros, intercepts, extrema, intervals of increase, and decrease, and rates of change .

Vertex Form of the Quadratic Determine the vertex of the following functions: • f(x) = 2(x – 1) 2 + 1 • g(x) = -(x + 3) 2 + 5 • h(x) = 3(x – 2) 2 – 7

Vertex & Axis of Symmetry Summary  Put equation in standard form f(x) = ax 2 + bx + c  Determine the value “a” and “b”  Determine if the graph opens up (a > 0) or down (a < 0)  Find the axis of symmetry:

x

 

b

2

a

 Find the vertex by substituting the “x” into the function and solving for “y”  Determine two more points on the same side of the axis of symmetry  Graph the axis of symmetry, vertex, & points

Vertex Form of the Quadratic Determine the vertex of the following equations: f(x) = 2(x – 1) 2 + 1 V = (1, 1) g(x) = -(x + 3) 2 + 5 h(x) = 3(x – 2) 2 – 7 V = (-3, 5) V = (2, -7) the same as the number added or subtracted at the end.

Vertex Form of the Quadratic  The vertex form of the quadratic equation is of the form:  y = a(x – h) 2 + k, where:  The vertex is located at (h, k)  The axis of symmetry is x = h  The “a” is the same as in the standard form  The “a” is the stretch of the function  The vertex is shifted right by h  The vertex is shifted up by k

Vertex Form of the Quadratic From y = x 2 Stretch factor Vertex Shift VERTICAL amount

y = a(x – h)

2

+ k

Vertex Shift HORIZONTAL amount

-6 6 5 f x   2 a > 0 Graph Opens UP Vertex is a MINIMUM 4 Parent Curve: 3 h x = x 2 2 1 g x   2 +5 a < 0 Graph Opens DOWN Vertex is a MAXIMUM 2 4 -4 -2 -1 6

In Class:  Do page 63 of Note Taking Guide  Do first 6 problems of Henley Task Day 2 – be sure to graph the y = x 2 for each graph.

In Class  Do page 64 of the Note Taking Guide  Do Day 2 of the Henley Task, # 4a – 4e all

Intercept Form of the Quadratic Function How can we determine the vertex of the following equations without putting them in standard form?

• f(x) = (x – 3)(x – 1) V = (2, -1) • g(x) = 2(x + 1)(x + 4) • h(x) = -3(x – 2)(x + 3) V = (-2.5, -4.5) V = (-0.5, 18.75) • Determine the x-intercepts (zero prod rule) • Find the axis of symmetry (average) • Find “y” value of the vertex (sub into f(x))

Homework  Page 65, # 1, 2, and 19 – 22 all

Convert from Standard to Vertex Form Standard: MM2A3a Students will Convert between standard and vertex form.

Convert from Standard to Vertex Forms  We converted from Vertex form to Standard form of the quadratic function above in slide 3 by expanding the (a – h) 2 term and combining like terms  How can we convert from Standard form to Vertex form?

Convert from Standard to Vertex Forms  Look at the standard form: y = ax 2 + bx + c, where a ≠ 0  And look at the Vertex form: y = a(x – h) 2 + k  “h” is the axis of symmetry, which is the “x” part of the coordinates of the vertex  “k” is the “y” part of the vertex

Convert from Standard to Vertex Forms  How did we find the axis of symmetry?



x

 

b

2

a

This is the “h” of the vertex form  How did we then find the “y” part of the vertex?

 Substitute the x into the original equation and solve for y.

 This is the “k” of the vertex form  The “a” is the same for both forms

Convert from Standard to Vertex Forms  Convert the following functions to vertex form:  f(x) = x 2 + 10x – 20  y = (x + 5) 2 - 45  g(x) = -3x 2 – 3x + 10  y = -3(x + 0.5) 2 + 10.75

 h(x) = 0.5x

2 – 4x – 3  y = 0.5(x – 4) 2 - 11