2. Quadratic functions can be
analyzed and represented in
three different ways: (a) using a
table of values, (b) graphically,
and (c) using the equation.
Let's explore each method:
3. (a) Table of Values: To create a
table of values for a quadratic
function, choose different values
for x and calculate the
corresponding y-values using
the quadratic equation. Here's
an example:
4. Consider the quadratic function
f(x) = 2x2 - 3x + 1.
Choose some x-values, say -2, -
1, 0, 1, and 2. Calculate the
corresponding y-values using
the equation: f(x) = 2x2 - 3x + 1.
5. Using the values selected, we
can generate a table like this:
x f(x)
-2 11
-1 6
0 1
1 0
2 1
6. (b) Graph: To graph a quadratic
function, plot points on a coordinate
plane using a table of values or by
recognizing important characteristics of
the function. In this case, we can plot
the points from the table or calculate
additional points if needed. Then,
connect the points to form a smooth
curve.
7. For the quadratic function
f(x) = 2x2 - 3x + 1, the graph will be
a parabola. The general shape and
direction of the parabola can be
determined by the coefficient "a" in
the quadratic equation. In this case,
since "a" is positive (a=2), the
parabola opens upward.
8. (c) Equation: The quadratic function
can also be represented using the
equation. In this case, the equation is
f(x) = 2x2 - 3x + 1. The equation
provides valuable information about the
function, such as the coefficients (a, b,
c) and key features like the vertex, axis
of symmetry, and the direction of
opening.
9. For example, using the quadratic
formula, we can find the vertex of the
parabola, which is given by: x = -b /
(2a) = -(-3) / (2 * 2) = 3/4
To find the corresponding y-value,
substitute x = 3/4 into the equation: f(3/4)
= 2(3/4)2 - 3(3/4) + 1 = 1/8
10. So, the vertex is (3/4, 1/8), and the axis
of symmetry is x = 3/4. By analyzing the
equation, we can also determine the y-
intercept, x-intercepts (if any), and the
minimum or maximum value achieved
by the function.
11. Remember, the equation, graph, and
table of values collectively provide a
comprehensive understanding of the
quadratic function.
13. (a) Using table of Values
Step 1: The equation of the quadratic function is f(x)
= 2x^2 - 3x + 1.
Step 2: Let's say we want to calculate the values for
x ranging from -2 to 2.
Step 3: Substitute each x-value into the equation to
find the corresponding y-values: - For x = -2: f(-2) =
2(-2)^2 - 3(-2) + 1 = 12 - For x = -1: f(-1) = 2(-1)^2 -
3(-1) + 1 = 6 - For x = 0: f(0) = 2(0)^2 - 3(0) + 1 = 1 -
For x = 1: f(1) = 2(1)^2 - 3(1) + 1 = 0 - For x = 2: f(2)
= 2(2)^2 - 3(2) + 1 = 3
Step 4: Create a table with two columns: | x | f(x) | |--
