This document contains instructions and examples for graphing quadratic functions. It begins with a review of key characteristics of quadratics such as their standard form and parabolic shape. The document then provides examples of graphing different types of quadratics by making tables of x-y values and plotting the points. It demonstrates how to find the axis of symmetry and vertex of a parabola by using the standard form equation. Students are instructed to graph various quadratics as class work.
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
1. Today:
Warm-Up
Review Quadratic Characteristics
Graphing Various Quadratic Functions
Class Work
2. Warm- Up Exercises
The slope is 2,
which is
positive
and the Y-
intercept
is -2
Therefore,
the
correct
graph is
A
3. Warm- Up Exercises
Write the equation for the line below:
The Y-intercept is: 0
The slope is: 2
The equation of the line is:
Y = 2x
4. Warm- Up Exercises
3. Write the inequality for the graph below
The Y-intercept is: 2
The slope is: -3
The line is solid,
not dotted. The
equation is:
Y < -3x + 2
5. Quadratic Review
A variable in a quadratic equation can have an exponent
of 2, but no higher.
The following are all examples of quadratic equations:
x2 = 25, 4y2 + 2y - 1 = 0, y2 + 6y = 0, x2 + 2x - 4 = 0
The standard form of a quadratic is written as:
ax2 + bx + c = 0, where only a cannot = 0
A. The graphs of
quadratics are not straight
lines, they are always in
the shape of a Parabola.
6. Graphing Parabolas & Parabola Terminology
Important points on a Parabola:
1.Axis of Symmetry:The axis of symmetry is the verticle
or horizontal line which runs through the exact center
of the parabola.
7. Graphing Parabolas & Parabola Terminology
Important points on a Parabola:
2. Vertex: The vertex is the highest point (the
maximum), or the lowest point (the minimum) on a
parabola.
Notice that the axis of
symmetry always runs
through the vertex.
8. Graphing Various Types of Quadratic Equations
Remember, the standard form of a quadratic equation is:
ax2 + bx + c = 0
Since the solutions or roots to a standard equation are
where the line crosses the x-axis, the y value is always
zero. As such, we can substitute y for zero:
y = ax2 + bx + c
Finally, since the y variable is dependent on the x, or is a
function of x, we can substitute the y for the function of
x, or (f)x:
(f)x = ax2 + bx + c
Regardless of which form is presented, the problem is
solved in the same way.
9. Graphing Various Types of Quadratic Equations
In this lesson, you will graph quadratic functions where b = 0.
The first step is to make a table. We can use the following x
values today:
Then complete the values for y and graph the parabola. This
must be done for each graph completed today.
11. Graphing Various Types of Quadratic Equations
Using the same graph, graph y = - ¼x2. Compare this graph
with the other two.
The first step is to make a table. We can use the following x
values :
15. Class Work:
Girls, do odd problems; Guys even.
Create tables for each problem.
One problem for each graph.
16. Finding the Axis of Symmetry and Vertex
1. Finding the Axis of Symmetry: The formula is: x = - b/2a
Plug in and solve for y = x2 + 12x + 32
We get - 12/2; = -6. The center of the parabola crosses the x
axis at -6. Since the axis of symmetry always runs through
the vertex, the x coordinate for the vertex is -6 also.
17. Finding the Axis of Symmetry and Vertex
y = x2 + 12x + 32
There is one more point left to find and that is the
y-coordinate of the vertex. To find this, plug in the
value of the x-coordinate back into the equation
and find y.
y = -62 + 12(-6) + 32. Y = 36 - 72 + 32; y = - 4
The bottom of the parabola is at -1 on the x axis, and
- 4 on the y axis.