Algebra 2
Chapter 7 Quadratic Equations and
Functions
7-5 Graphing y – k = a(x – h)2
 WARMUP:
 Create a table of (x, y) values for the
1.
2.
3.
4.
following functions:
3x + 4y = 12
x – 2y = 8
y = |x|
Now GRAPH #3 from above.
7-5 Graphing y – k = a(x – h)2
 GOAL: To graph parabolas whose
equations have the form y-k = a(x-h)2,
and to find the vertices and the axes of
symmetry.
7-5 Graphing y – k = a(x – h)2
 Let’s make a table of values and then
graph.
yx
2
7-5 Graphing y – k = a(x – h)2
 This resulting curve is called a
parabola.
 Notice that points (x, y) and (-x, y) are
“mirror images” of each other across
the y-axis.
 Because of this, the y-axis is called the
axis of symmetry, or just axis, of this
parabola.
7-5 Graphing y – k = a(x – h)2
 The vertex of the parabola is where the
parabola crosses its axis.
 For y=x2, the vertex is the origin.
7-5 Graphing y – k = a(x – h)2
 Now let’s examine how small changes
affect what the graph looks like:
 How about y = -x2?
7-5 Graphing y – k = a(x – h)2
 How about a slightly more generic
y=ax2?
 y = 3x2
 y = (1/2)x2
7-5 Graphing y – k = a(x – h)2
 The graph of y=ax2 opens upward if
a>0 and downward if a<0.
 The larger |a| is, the “narrower” the
graph is.
7-5 Graphing y – k = a(x – h)2
 Now what about y = a(x – h)2?
 y = 1(x – 0)2
 y = 1(x – 3)2
 y = 1(x – (-3))2
7-5 Graphing y – k = a(x – h)2
 To graph y = a(x – h)2, slide the graph
of y = ax2 horizontally h units.
If h>0, slide to the right.
If h<0, slide to the left.
The graph has vertex (h, 0) and its axis
is the line x=h.
7-5 Graphing y – k = a(x – h)2
 Next, we have y – k = ax2.
 y – 3 = 1x2
 Y + 3 = 1x2
7-5 Graphing y – k = a(x – h)2
 To graph y – k = ax2, slide the graph of
y = ax2 vertically k units.
If k>0, slide it upward; if k<0, slide it
downward.
The graph has vertex (0, k) and its
axis is the line x = 0 (the y-axis).
7-5 Graphing y – k = a(x – h)2
 Now put them all together.
7-5 Graphing y – k = a(x – h)2
 To graph y – k = a(x – h)2,
slide the graph of y=ax2 horizontally
h units and vertically k units.
The graph has a vertex (h, k) and its
axis is the line x = h.
7-5 Graphing y – k = a(x – h)2
 Examples
7-5 Graphing y – k = a(x – h)2
 The y-coordinate of a point where the
graph crosses the y-axis is called the
y-intercept. The x-coordinate of a point
where a graph crosses the x-axis is
called the x-intercept.
 A parabola may have no x-intercepts,
one x-intercept or two x-intercepts.
7-5 Graphing y – k = a(x – h)2
 Examples of intercepts…
7-5 Graphing y – k = a(x – h)2
 To find the y-intercept of a parabola,
set x equal to zero and solve for y.
 To find the x-intercepts of a parabola,
set y equal to zero in the equation and
solve the resulting quadratic equation
for x. If the roots are real, they are the
x-intercepts. If the roots are imaginary,
then the graph has NO x-intercepts.
7-5 Graphing y – k = a(x – h)2
 Graph y + 6 = 2(x + 1)2. Label the
vertex, axis and find all intercepts.
7-5 Graphing y – k = a(x – h)2
 Find an equation in y – k = a(x – h)2
form with:
vertex is (4, 5) and contains (5, 3)
7-5 Graphing y – k = a(x – h)2
 More examples?
7-5 Graphing y – k = a(x – h)2
 HOMEWORK!!