Deconstructing the Graph of a Quadratic Function (3.14.1)

Question 1

• Quadratic functions:

1) represent parabolas;
2) have symmetry;
3) have a turning point, either a maximum or a minimum.

Answer 1

• Quadratic functions:

1) represent parabolas;
2) have symmetry;
3) have a turning point, either a maximum or a minimum.

Question 2

• The domain for quadratic functions is the set of all real numbers.

Answer 2

• The domain for quadratic functions is the set of all real numbers.

Question 3

Your function is f (x) = (x +3)2
– 4, so you know that your
vertex is not going to be at the origin.
One way to graph a parabola is to randomly choose values
for x, substitute them into the expression, calculate the
resulting y-value, and plot each point.
If you do that for this example, you will get this graph.
Here’s another approach you can use either to create your
graph or to check your arithmetic.
Step One: You know that all quadratics graph a parabola.
Here is the standard “happy” parabola with its vertex at the
origin.
Step Two: Check for any shift away from the origin.
Be careful about shifting from the origin that you shift both
sideways and vertically in the correct direction.
Here’s Dr. Burger’s rule for shifting. Learn this rule to be
sure you always shift in the correct direction.
Checking your function, f (x) = (x +3)2
– 4, you see that you
are directed to shift to the left 3 units and down 4 units from
the origin.
You can do that and locate your graph.
In this example, your domain is the set of all real numbers.
Your range is more limited. The smallest y-value that you
will have is the one at the vertex, which is –4. As a result,
your range is limited to all the numbers between –4 and +∞.

Answer 3

Your function is f (x) = (x +3)2
– 4, so you know that your
vertex is not going to be at the origin.
One way to graph a parabola is to randomly choose values
for x, substitute them into the expression, calculate the
resulting y-value, and plot each point.
If you do that for this example, you will get this graph.
Here’s another approach you can use either to create your
graph or to check your arithmetic.
Step One: You know that all quadratics graph a parabola.
Here is the standard “happy” parabola with its vertex at the
origin.
Step Two: Check for any shift away from the origin.
Be careful about shifting from the origin that you shift both
sideways and vertically in the correct direction.
Here’s Dr. Burger’s rule for shifting. Learn this rule to be
sure you always shift in the correct direction.
Checking your function, f (x) = (x +3)2
– 4, you see that you
are directed to shift to the left 3 units and down 4 units from
the origin.
You can do that and locate your graph.
In this example, your domain is the set of all real numbers.
Your range is more limited. The smallest y-value that you
will have is the one at the vertex, which is –4. As a result,
your range is limited to all the numbers between –4 and +∞.