 ## Presentation on theme: "Graphs of Quadratic Functions"— Presentation transcript:

Quadratic Functions: Definition of Quadratic Function: The graph of a quadratic function is a special type of “U”-shaped curve that is called a parabola. (U shaped) All parabolas are symmetric with respect to a line called the axis of symmetry, or simply the axis of the parabola. The point where the axis intersects the parabola is the vertex of the parabola.

Quadratic Equations y = ax2 + bx + c y = a(x - h)2 + k f(x) =
Standard form Vertex form (later) The graph is “U-shaped” and is called a parabola.

The highest or lowest point on the parabola is called the vertex.

The axis of symmetry for the parabola is the vertical line through the vertex.

y = x2 y = -x2 Vertex is a minimum Vertex is a maximum Opens up
Opens down

Finding the vertex of a parabola:
This gives you the x-value and plug the x-value into the original function to find the y-value

EXAMPLES: Sketch the graph of the quadratic function without using a graphing utility. Identify the vertex and x-intercepts. You know the graph will go up b/c a is a positive number Find the vertex first!! So the x part of the vertex is -2 Now, to find the y-value, plug in -2 wherever you see an x. Vertex: (-2,-7)

Now make the table! Remember the vertex goes in the middle!!! (x, y) x
-2

Sketch the graph of the quadratic function without using a graphing
EXAMPLES: Sketch the graph of the quadratic function without using a graphing utility. Identify the vertex and x-intercepts. You know the graph will go up b/c a is a positive number Find the vertex first!! Where is B??? So the x part of the vertex is 0 Now, to find the y-value, plug in zero wherever you see an x. Vertex: (0,-5)

So, we know that the vertex is (0,-5) now all we have to do is a
T-chart and pick 4 more points and then graph. x y 0 -5

One More…. Name the vertex, axis of symmetry and whether it opens up or down AND GRAPH. y = 2x2 + 1 Vertex: (0, 1) a = 2 b = 0 c = 1 Axis of sym: x = 0 Opens up

Name the vertex, axis of symmetry and whether it opens up or down
AND GRAPH. y = 2x2 + 1 Vertex: (0, 1) x y Put vertex in the middle of the t-table -2 9 -1 3 1 1 3 2 9

Example Graph the quadratic function f (x) = -x2 + 6x -. Solution:
Step 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 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 x-coordinate: 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: the parabola has its vertex at (3,7).

Example Graph the quadratic function f (x) = -x2 + 6x -.
Step Find the x-intercepts. Replace f (x) with 0 in f (x) = -x2 + 6x = -x2 + 6x - 2

Example Graph the quadratic function f (x) = -x2 + 6x -.
Step Find the y-intercept. Replace x with 0 in f (x) = -x2 + 6x - 2. f (0) = • = - The y-intercept is –2. The parabola passes through (0, -2). Step Graph the parabola.