1. Quadratic Functions and Their Graphs
More in Sec. 2.1b
Homework: p odd

2. What are they??? Quadratic Function – a polynomial function of
degree 2
Recall the basic squaring function?
 Any quadratic function can be obtained via a
sequence of transformations of this basic
function…………observe

3. reflection across x-axis,
Quick Examples
Describe how to transform the basic squaring function into the
graph of the given function. Sketch its graph by hand.
Vertical shrink by 1/2,
reflection across x-axis,
translation up 3 units

4. Translation left 2 units,
Quick Examples
Describe how to transform the basic squaring function into the
graph of the given function. Sketch its graph by hand.
Translation left 2 units,
vertical stretch by 3,
translation down 1 unit

5. More Generalizations…
Consider the graph of
If , the parabola
opens downward
If , the parabola
opens upward
Axis of Symmetry (axis for short) – line of symmetry
Vertex – point where the parabola intersects the axis

6. Definition: Vertex Form of a Quadratic Function
(Standard Quadratic Form)
Any quadratic function , , can be
written in the vertex form
The graph of f is a parabola with vertex (h, k ) and axis x = h,
where and If a > 0, the parabola
opens upward, and if a < 0, it opens downward.

7. Guided Practice Vertex: (–2, 5) Axis: x = –2 Vertex: (3/2, –1)
Find the vertex and axis of the graph of the given functions.
Vertex: (–2, 5)
Axis: x = –2
Vertex: (3/2, –1)
Axis: x = 3/2

8. Guided Practice
Use vertex form of a quadratic function to find the vertex and
axis of the given function. Rewrite the equation in vertex form.
Standard form:
So, a = –3, b = 6, and c = –5
Coordinates of the vertex:

9. How about a graph to support these answers?
Guided Practice
Use vertex form of a quadratic function to find the vertex and
axis of the given function. Rewrite the equation in vertex form.
Vertex:
Axis:
Vertex form of f :
How about a graph to support these answers?

10. First, let’s make sure we remember how to complete the square…
Solve by completing the square:
Get x terms by themselves
Complete the square!!!
Factor
Take square root of both sides
Solve for x

11. We can complete a similar process when changing forms of quadratics:
Use completing the square to describe the graph of the
given function. Support your answer graphically.
The graph of f is a upward-opening parabola
with vertex (–2, –1), axis x = –2, and intersects
the x-axis at about –2.577 and –1.423.

12. Characterizing the Nature of a Quadratic Function
Point of View
Characterization
Verbal
Polynomial of degree 2
Algebraic or
Graphical
Parabola with vertex (h, k), axis x = h;
opens upward if a > 0, opens downward
if a < 0; initial value = y-int = f(0) = c;
x-intercepts:

13. Guided Practice Vertex: (5/2, –77/4), Axis: x = 5/2, Opens
Use completing the square to describe the graph of the
given function. Support your answer graphically.
Vertex: (5/2, –77/4), Axis: x = 5/2, Opens
upward, intersects the x-axis at about 0.538
and 4.462, Vertically stretched by 5.

14. Guided Practice Check with a calculator graph!!!
Write an equation for the parabola shown, using the fact
that one of the given points is the vertex.
Plug in (3, –2) for (h, k):
(6, 1)
Plug in (6, 1) for (x, y), solve for a:
(3, –2)
Check with a
calculator graph!!!

15. Guided Practice Check with a calculator graph!!!
Write an equation for the parabola shown, using the fact
that one of the given points is the vertex.
Plug in (–1, 5) for (h, k):
(–1, 5)
Plug in (2,–13) for (x, y), solve for a:
(2,–13)
Check with a
calculator graph!!!

16. Guided Practice Check with a calculator graph!!!
Write an equation for the quadratic function whose graph
contains the vertex (–2, –5) and the point (–4, –27).
Plug in the vertex:
Plug in the point:
Check with a
calculator graph!!!