Download notes 2 Standard Form File

Chapter 6: Quadratic Functions
Math 2201
Handout #2
Standard Form
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Standard form for the equation of a quadratic function is 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
where a, b, and c are real numbers and a cannot be equal to zero
It can also be written using function notation 𝑓(𝑥) = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
Determining the Properties of the Function Written in Standard Form
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All properties are determined based on the information in handout #1
1.
Vertex**
• The vertex of a quadratic function is the turning point of the graph.
• Determine the x-coordinate: 𝒙 = − 𝒃
𝟐𝒂
• Determine the y-coordinate: substitute the x-coordinate into the equation and
solve for y
2.
Direction of Opening
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3.
A quadratic function can open Up or Down
It is determined by the sign of the leading coefficient (the value of “a”)
If “a” is POSITIVE ( a > 0 ) then the graph opens UP
If “a” is NEGATIVE ( a < 0 ) then the graph opens DOWN
Maximum or Minimum Point
• The vertex is always the maximum or minimum point on the graph of a quadratic
function. It is one OR the other. It can’t be both
• The vertex is the MINIMUM point (Lowest) point on the graph when the graph OPENS
UP. It is on the bottom of the graph. This happens when “a” is positive ( a > 0 )
• The vertex is the MAXIMUM point (Highest) point on the graph when the graph
OPENS DOWN. It is at the top of the graph. This happens when “a” is negative ( a < 0 )
4.
Maximum or Minimum Value
• The y-coordinate of the vertex is the highest or lowest value for y for the function
• It is the MINIMUM value when the vertex is at the bottom of the graph, when the
graph OPENS UP. This happens when “a” is positive ( a > 0 )
• It is the MAXIMUM value when the vertex is at the top of the graph, when the graph
OPENS DOWN. This happens when “a” is negative ( a < 0 )
5.
Width of the Graph
• The graph can be wider or thinner or the same width as the original graph of y = x2 .
This is determined by the number value (ignore the sign) of the leading coefficient, “a”.
• The graph is THINNER than y = x2 when the value of “a” is higher than 1 (a > 1)
• The graph is THE SAME width as y = x2 when the value of “a” is equal to 1 (a = 1)
• The graph is WIDER than y = x2 when the value of “a” is between 0 and 1
(0 < a < 1)
6.
Y-intercept
• The y-intercept of any graph is where the graph crosses the y-axis. It is the answer for
y when the value of x is zero
• Depending on the form of the equation, the y-intercept can be determined from the
equation. In standard form the y-intercept is “c” the constant number at the end
7.
Axis (Line) of Symmetry
• The line (or Axis) of symmetry for any parabola is the vertical line that is drawn down
through the vertex.
• The line of symmetry divides the graph into two equal parts that are mirror images of
each other.
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Chapter 6: Quadratic Functions
Math 2201
• Has an equation 𝑥 = (𝑡ℎ𝑒 𝑥 − 𝑐𝑜𝑜𝑟𝑑𝑖𝑛𝑎𝑡𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑒𝑟𝑡𝑒𝑥)
8.
Handout #2
Domain
• The domain is the x-values that are allowed to be used in the function. Any number
can be put into the equation to get an answer for y.
• The domain is All Real Numbers 𝑥 ∈ 𝑅
9.
Range
• The range is the y-values that are possible based on the domain values. The range
depends on (i) the direction of opening (determined by the sign of “a”) and (ii) the ycoordinate of the vertex (k)
• When the graph OPENS UP (a > 0) the range is the y-coordinate of the vertex and all
the y-values above it. It is written y  k where k is the y-coordinate of the vertex
• When the graph OPENS DOWN (a < 0) the range is the y-coordinate of the vertex and
all the y-values below it. It is written y  k where k is the y-coordinate of the vertex
10.
To Draw the Graph
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Determine and plot the coordinates of the vertex
Draw the line of symmetry through the vertex
Determine and plot the y-intercept; plot its mirror image across the line of symmetry
Choose another x-value and find its y-value. Plot this point and its mirror image across
the line of symmetry
• If the vertex is on the y-axis then it is the y-intercept as well. To find a second pair of
points, choose another x-value, find its y-coordinate. Plot this point and its mirror
image
(a)
(b)
𝑦 = 𝑥 2 − 8𝑥 + 6
(c)
𝑦 = −2𝑥 2 + 4𝑥 − 1
1
𝑦 = 2 𝑥 2 + 2𝑥 − 4
Direction of Opening
Maximum/Minimum
Point
Width of Graph
Vertex
Axis of Symmetry
Y-intercept
Maximum/Minimum
Value
Domain
Range
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