Algebra (Period 1)

Linear Functions 12-3 and Quadratic Functions: 12- 4

Linear Functions

y = mx +b is a linear function.
f(x) = mx + b

graph it-- we always get a straight line

Real Life problems

Rental Truck Co charges $35 a day plus $.21 per mile. Find the cost of renting a truck for a day trip of 340 miles.

f(x) = .21m + 35
f(340) = .21(340) + 35
f(340) = 106.50
$106.50

The cost of renting a chain saw is $5.90 a hour plus $6.50 for a can of gas. Find the cost of using the chain saw for 7.5 hours.

f(h) = 5.90h + 6.50
f(7.5) = 5.90(7.5) = 6.50
f(7.5) = 50.75
$50.75

Quadratic Functions: 12- 4
A QUADRATIC FUNCTION is not y = mx + b
(which is a LINEAR function),
but instead is
y = ax² + bx + c
OR
f(x) = ax² + bx + c
where a, b, and c are all real numbers and
a cannot be equal to zero because
it must have a variable that is squared ( degree of 2)
[If a = 0, then we would end up with y = bx + c which is really y= mx + b]

Quadratics have a squared term, so they have TWO possible solutions also called roots. You already saw this in Chapter 6 when you factored the trinomial and used zero products property!! ( CHAPTER 6-- again)

If the domain is all real numbers, then you will have a PARABOLA which looks like
a smile when the a coefficient is positive or
looks like a frown when the a coefficient is negative.


How is the graph of y = 2x² related to the graph of y = x²?
How is the graph of y = 2x² related to the graph of y =-2x²?

The vertex is the maximum or minimum point of a parabola. It is the maximum point with a quadratic such as y = -x²
and it is a minimum point with a quadratic in the form y = x²
Axis of symmetry- if you fold the graph so the two sides of the parabola coincide, the the fold line is the axis of symmetry.
THe y-axis is the axis of symmetry for all equations of the form y = ax².

For a parabola defined by the equation y = ax² + bx + c
the x coordinate of the vertex is -b/2a
and the line of symmetry is x = -b/2a





Graphing quadratics:
You can graph quadratics exactly the way you graphed lines
by plugging in your choice of an x value and using the equation to find your y value.

Because it's a U shape, you should graph 5 points as follows:
STEP 1: determine Point 1: the vertex -
the minimum value of the smile or
the maximum value of the frown

The x value of the VERTEX = -b/2a
We get the values for a and b from the actual equation
f(x) = ax² + bx + c
just plug in the b and the a value from your equation into -b/2a and you have the x-value of the vertex.
Now to find the y value -- take that x value and PLUG it into the equation


STEP 2: Next, draw the AXIS OF SYMMETRY : x = -b/2a
a line through the vertex parallel to the y axis . Draw this line as a dashed line. REMEMBER: It will be a dashed line parallel to the y-axis


STEP 3: Point 2- Pick an x value to the right or left of the axis and find its y by plugging into the equation.

STEP 4: Point 3- Graph its mirror image on the other side of the axis of symmetry by counting from axis of symmetry

STEP 5: Points 4 and 5- Repeat point 2 and 3 directions with another point even farther from the vertex

JOIN YOUR 5 POINTS IN A "U" SHAPE AND EXTEND LINES WITH ARROWS ON END

Parabolas that are functions have domains that are ALL REAL NUMBERS
Their ranges depend on where the vertex is and also if the a coefficient is positive or negative

EXAMPLE: f(x) = -3x² (or y = -3x²)
the a coefficient is negative so it is a frown face
the x value of the vertex (maximum) is -b/2a or 0/2(-3) = 0
the y value of the vertex is 0
So the vertex is (0, 0)


To graph this function:
1) graph vertex (0, 0)
2) Draw dotted line x = 0 (actually this is the y axis!)
3) Pick x value to the right of axis of symmetry, say x = 1
Plug it in the equation: y = -3(1) = -3
Plot (1, -3)
4) count steps from axis of symmetry and place another point to the LEFT of axis in same place
5) pick another x value to the right of the axis of symmetry, say x = 2
plug it in the equation y = -3(2) = -6
plot ( 2, -6). Count the steps from the axis of symmetry and place another point to the LEFT of the axis in the same place.

The domain is all real numbers.
The range is y is less than or equal to zero