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Vertex and Intercept Form of Quadratic Function

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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

- Put equation in standard form f(x) = ax2 + 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:
- 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

V = (1, 1)

V = (-3, 5)

V = (2, -7)

Determine the vertex of the following equations:

f(x) = 2(x – 1)2 + 1

g(x) = -(x + 3)2 + 5

h(x) = 3(x – 2)2 – 7

Compare the equations and the vertices. Do you notice a pattern?

The x part is the opposite sign of the number inside the brackets and the y part is the same as the number added or subtracted at the end.

- 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

From y = x2

Stretch factor Vertex Shift VERTICAL amount

y = a(x – h)2 + k

Vertex Shift HORIZONTAL amount

- Do page 63 of Note Taking Guide
- Do first 6 problems of Henley Task Day 2 – be sure to graph the y = x2 for each graph.

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

Intercept Form of the Quadratic Function

V = (2, -1)

V = (-2.5, -4.5)

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

f(x) = (x – 3)(x – 1)

g(x) = 2(x + 1)(x + 4)

h(x) = -3(x – 2)(x + 3)

Determine the x-intercepts (zero prod rule)

Find the axis of symmetry (average)

Find “y” value of the vertex (sub into f(x))

V = (-0.5, 18.75)

- Page 65, # 1, 2, and 19 – 22 all

- Standard: MM2A3a Students will
- Convert between standard and vertex form.

- 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?

- Look at the standard form:
y = ax2 + 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

- How did we find the axis of symmetry?
- 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 the following functions to vertex form:
- f(x) = x2 + 10x – 20
- y = (x + 5)2 - 45
- g(x) = -3x2 – 3x + 10
- y = -3(x + 0.5)2 + 10.75
- h(x) = 0.5x2 – 4x – 3
- y = 0.5(x – 4)2 - 11