Graphing Quadratic Functions
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Transcript Graphing Quadratic Functions
Graphing Quadratic Functions
y=
2
ax
+ bx + c
Graphing Quadratic Functions
Today we will:
Understand how the coefficients of a quadratic
function influence its graph
a.
b.
c.
d.
e.
The direction it opens (up or down)
Its vertex
Its line of symmetry
Its y-intercepts
Its x-intercepts (roots)
Key terms
• Parabola – the graph of a
quadratic function
• Quadratic function – a
function that can be
written in the form
y ax2 bx c, where a 0.
• Standard form of a quadratic function is:
y ax2 bx c, where a 0.
Key terms - continued
• Vertex – the point where a parabola
crosses its line of symmetry.
• Maximum – the vertex of a parabola that
opens downward. The y-coordinate of the
vertex is the maximum value of the
function.
• Minimum – the vertex of a parabola that
opens upward. The y-coordinate of the
vertex is the minimum value of the
function.
• y-intercept – the y-coordinate of the
point where a graph crosses the y-axis.
• x-intercept – the x-coordinate of the point
where a graph crosses the x-axis.
I
Line of symmetry
Line of symmetry
Parabola
y
The graph of a quadratic function is a
parabola.
Vertex
A parabola can open up or down.
If the parabola opens up, the lowest
point is called the vertex.
x
If the parabola opens down, the
vertex is the highest point.
Vertex
Finding the Vertex
We know the line of symmetry
always goes through the vertex.
Thus, the line of symmetry gives
us the x – coordinate of the
vertex.
To find the y – coordinate of the
vertex, we need to plug the x – value
into the original equation.
y = –2x2 + 8x –3
STEP 1: Find the line of symmetry
x b 8 8 2
2a 2(2) 4
STEP 2: Plug the x – value into the original
equation to find the y value.
y = –2(2)2 + 8(2) –3
y = –2(4)+ 8(2) –3
y = –8+ 16 –3
y=5
Therefore, the vertex is (2 , 5)
Vertex : Max & Min
y
The standard form of a quadratic
function is
a>0
y = ax2 + bx + c
The parabola will open up when
the a value is positive.
x
The parabola will open down
when the a value is negative.
a<0
Minimum and Maximum: Quadratic
Functions
•
Consider f(x) = ax2 + bx +c.
1. If a > 0, then f has a minimum that occurs at
x = -b/(2a). This minimum value is f(-b/(2a)).
2. If a < 0, the f has a maximum that occurs at
x = -b/(2a). This maximum value is f(-b/(2a)).
Line of Symmetry
Parabolas have a symmetric
property to them.
Line yof
Symmetry
If we drew a line down the middle
of the parabola, we could fold the
parabola in half.
We call this line the line of
symmetry.
Or, if we graphed one side of the
parabola, we could “fold” (or
REFLECT) it over, the line of
symmetry to graph the other side.
x
The line of symmetry ALWAYS passes
through the vertex.
Finding the Line of Symmetry
When a quadratic function is in
standard form
y=
ax2
+ bx + c,
The equation of the line of symmetry
is
x b
2a
For example…
Find the line of symmetry of y
= 3x2 – 18x + 7
Using the formula…
x 18 18 3
2 3 6
This is best read as …
the opposite of b divided by the
quantity of 2 times a.
Thus, the line of symmetry is x = 3.
Graphing Parabolas With Equations
in Standard Form
To graph f (x) y = ax2 + bx + c:
1. Determine whether the parabola opens upward or
downward. If a > 0, it opens upward. If a < 0, it opens
downward.
2. Determine the vertex of the parabola.
3. Find any x-intercepts by replacing f (x) with 0. Solve the
resulting quadratic equation for x. (Solving the
functions when y = 0 )
4. Find the y-intercept by replacing x with zero.
5. Plot the intercepts and vertex. Connect these points
with a smooth curve that is shaped like a cup.
Example 1
y
2
x
3
x
1
Use the function
2
A. Tell whether the graph opens up or down.
B. Tell whether the vertex is a maximum or a
minimum.
C. Find an equation for the line of symmetry.
D. Find the coordinates of the vertex.
Example 1 Solution
Use the function y 2 x2 3x 1
A. a is positive, so the graph opens up.
B. The vertex is a minimum.
C. Equation for the line of symmetry.
3
3
x
2(2)
4
D. Coordinates of the vertex.
2
9 9
17
1
3
3
y 2 3 1 1 2
8 4
8
8
4
4
1
3
,
2
8
4
Example 2
Use the quadratic function y 3x 18x 25
2
A.
B.
C.
D.
E.
F.
Will the graph open up or down?
Is the vertex a minimum or a maximum?
What is the equation of the line of symmetry?
Find the coordinates of the vertex of the graph.
Find the y-intercept.
Graph the function.
Example 2 Solution
Use the quadratic function y 3x2 18x 25
A. The graph will open up, a is positive.
B. The vertex a minimum.
C. Equation of the line of symmetry.
b
18
x
3
2a
2(3)
D. Coordinates of the vertex of the graph.
y 3(3)2 18(3) 25 y 2
(3, 2)
E. The y-intercept is y = 25.
F. Graph the function.
Example 2 Solution
y 3x 18x 25
2
Use the quadratic function
F. Graph the function.
50
40
30
y = 25
20
line of symmetry
x=3
10
0
-2
0
2
4
-10
I
6
8
Example 3
Graph the quadratic function f (x) x 6x .
2
Solution:
Step 1 Determine how the parabola opens. Note that a, the coefficient of
x 2, is -1. Thus, a < 0; this negative value tells us that the parabola opens
downward.
Step 2 Find the vertex. We know the x-coordinate of the vertex is –b/2a.
We identify a, b, and c to substitute the values into the equation for the xcoordinate:
x = -b/(2a) = -6/2(-1) = 3.
The x-coordinate of the vertex is 3. We substitute 3 for x in the equation of
the function to find the y-coordinate:
y f (3) 32 6(3) 2 9 18 2 7
the parabola has its vertex at (3,7).
Example 3
Graph the quadratic function f (x) x2 6x .
Step 3
Find the x-intercepts. Replace f(x) with 0 in f(x) x2 6x 2.
0 = x2 6x 2
a 1,b 6,c 2
b b 2 4ac
x
2a
2
6 6 4(1)(2)
2(1)
6 36 8
2
6 28 6 2 7
2
2
3 7
Example 3
Graph the quadratic function f (x) x2 6x .
Find the y-intercept. Replace x with 0 in f(x) x2 6x 2.
f0 02 6 • 0 2
The y-intercept is –2. The parabola passes through (0, 2).
Step 5 Graph the parabola.
10
Step 4
8
6
4
2
-10 -8 -6 -4 -2
2
-2
-4
-6
-8
-10
4
6
8 10