Completing the Square
Complete the square for x2+6x. What do you add to make a perfect square trinomial?
Add 9 (or write as (x+3)2−9)
What is the vertex form of a quadratic? Write the general equation.
y=a(x−h)2+k
In the equation f(x)=(x−4)2, how does the graph shift compared to y=x2?
Shifts 4 units to the right
Use the formula x=−b/2a to find the x-coordinate of the vertex for y=x2+4x+1
x=−2
If a parabola has vertex (1, -3) and passes through (2, -1), write the equation in vertex form.
f(x)=2(x−1)2−3
When completing the square for x2−8x, what constant term do you add?
Add 16
In vertex form f(x)=2(x−3)2+5, what is the vertex?
(3,5)
Describe the transformation from y=x2 to y=2(x−1)2+3
Shift 1 right, 3 up, and stretch vertically by 2
Find the vertex of f(x)=2x2−12x+5
Vertex: (3, -13)
Convert 3x−12x+7 to vertex form. Then state the axis of symmetry.
Vertex form: 3(x−2)2−5; Axis of symmetry: x=2
Complete the square: x2+10x+5=0. Rewrite in the form (x+a)2=b.
(x+5)2=20
Convert x2+4x+7to vertex form.
(x+2)2+3
If a=−1 in vertex form, what does this tell you about the parabola?
The parabola opens downward (reflects over x-axis)
Given y=x2−2x−3, find the vertex and determine if it's a maximum or minimum.
Vertex: (1, -4); Minimum
A parabola with discriminant 0 has vertex at (3, 0). Write its equation in vertex form if a=1
f(x)=(x−3)2
Complete the square for 2x2+8x+3 Factor out the 2 first, then complete the square.
2(x+2)2−5
Write −3(x+1)2 -2 in vertex form and identify the vertex.
Vertex form: −3(x+1)2−2; Vertex: (-1, -2)
Compare f(x)=(x+2)2−4 and g(x)=0.5(x+2)2−4. How do the graphs differ?
g(x) is compressed vertically (narrower) compared to f(x)
Use completing the square to find the vertex of f(x)=−2x2+8x−5.
Vertex: (2, 3)
Find the zeros of f(x)=(x−2)2−9 by first converting to standard form, then solving.
x = -1 and x = 5
Complete the square: 3x2−12x+1. Show your work step-by-step.
3(x−2)2−11
Convert x2−6x+8 to vertex form and identify the minimum value.
(x−3)2−1; Minimum value: -1
Write the vertex form of a parabola that opens downward, has vertex at (-2, 3), and is vertically stretched by a factor of 3.
f(x)=−3(x+2)2+3
A quadratic has x-intercepts at 2 and 6. What is the x-coordinate of the vertex?
x=4 (midpoint between the roots)
A quadratic in vertex form is f(x)=−0.5(x+3)2+8. Identify the vertex, axis of symmetry, and direction of opening. Then find the y-intercept.
Vertex: (-3, 8); Axis: x=−3; Opens downward; Y-intercept: (0, -0.5)