Graphing Quadratics with a Table
Yesterday, we reviewed functions, and today we're diving specifically into quadratic functions. Remember that quadratic functions are based on the parent function $f(x) = x^2$. The graph of a quadratic function is a U-shaped curve called a parabola.
Understanding Quadratic Functions
A quadratic function can be written in the form:
$$f(x) = a(x - h)^2 + k$$where $a \neq 0$. Key vocabulary includes quadratic function and parabola. Understanding the parent function is crucial for graphing and recognizing transformations.
Graphing Using a Table
One way to graph a quadratic function is by creating a table of values. Let's consider the example:
$$f(x) = x^2 - 4x + 3$$To graph this, we can choose several $x$ values and calculate the corresponding $f(x)$ values:
- When $x = 0$, $f(0) = (0)^2 - 4(0) + 3 = 3$, so the point is $(0, 3)$.
- When $x = 1$, $f(1) = (1)^2 - 4(1) + 3 = 0$, so the point is $(1, 0)$.
- When $x = 2$, $f(2) = (2)^2 - 4(2) + 3 = -1$, so the point is $(2, -1)$.
- When $x = 3$, $f(3) = (3)^2 - 4(3) + 3 = 0$, so the point is $(3, 0)$.
- When $x = 4$, $f(4) = (4)^2 - 4(4) + 3 = 3$, so the point is $(4, 3)$.
By plotting these points and connecting them with a smooth curve, we can graph the quadratic function. Remember to choose enough points to see the entire curve!
Transformations of Quadratic Functions
Understanding transformations can make graphing easier. The general form $f(x) = a(x - h)^2 + k$ reveals how the parent function $f(x) = x^2$ is transformed:
- $a$: Vertical stretch (if $|a| > 1$) or compression (if $0 < |a| < 1$). If $a < 0$, the parabola is reflected across the x-axis.
- $h$: Horizontal shift. A positive $h$ shifts the graph to the right, and a negative $h$ shifts it to the left.
- $k$: Vertical shift. A positive $k$ shifts the graph upward, and a negative $k$ shifts it downward.
Practice Problems
Try graphing these functions using a table:
- $f(x) = x^2 + 2x + 1$
- $g(x) = -x^2 + 4$
Also, consider the following problems from your textbook:
- Pg. 64 # 2-4, 17-19
Real-World Examples and Discussion
The parabola is an important shape in our everyday world. For your discussion question, find a picture that shows an example of a parabola in the real world (not a textbook graph). Post a link to the image online. Remember, the first unique image gets the credit!