Lesson 6.2: Quadratic Functions and Their Graphs
Introduction
In this lesson, we will explore quadratic functions and their graphs. Quadratic functions are fundamental in mathematics and have a wide range of applications in various fields, including physics, engineering, and economics. This lesson aims to provide you, students, with an understanding of the following:
- The graph of $y = x^2$ and the general shape of a parabola.
- How to plot a quadratic function from a table of values.
- Identifying the turning point and line of symmetry of a parabola.
- Reading the roots (x-intercepts) of a quadratic function from its graph.
Hook
Imagine you throw a ball in the air. It follows a path that can be represented by a parabola—a U-shaped curve that is characteristic of quadratic functions. Understanding the fundamentals of these functions can help us predict the path of the ball, the height it will reach, and when it will hit the ground.
What is a Quadratic Function?
A quadratic function is a polynomial function of degree two, usually written in the standard form:
$$
egin{align*}
f(x) &= ax^2 + bx + c,
$\end{align*}$
$$
where:
- $a$, $b$, and $c$ are constants, and a
eq 0$ (if $a = 0, the function becomes linear).
- The coefficient $a$ determines the direction in which the parabola opens: if $a > 0$, it opens upwards; if $a < 0$, it opens downwards.
Graph of $y = x^2$
The simplest quadratic function is $y = x^2$. The graph of this function forms a parabola that opens upwards with its vertex at the origin (0,0).
To visualize the graph, we can create a table of values:
| $x$ | $y = x^2$ |
|---|---|
| -3 | 9 |
| -2 | 4 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
Plotting these points on a graph results in a U-shaped curve (parabola):
Worked Example: Plotting $y = x^2$
- Create a table of values for $x$ ranging from -3 to 3.
- Calculate corresponding $y$ values using $y = x^2$.
- Plot the points: (-3,9), (-2,4), (-1,1), (0,0), (1,1), (2,4), and (3,9).
- Connect the points smoothly to form the parabola.
Through this process, we observe that the graph is symmetric about the $y$-axis, confirming it has a line of symmetry along $x = 0$.
Identifying Turning Points and the Line of Symmetry
The turning point of a quadratic function is the minimum or maximum point on the graph, depending on whether the parabola opens upwards or downwards. For the function $y = x^2$, the turning point is at the vertex (0,0).
Line of Symmetry
The line of symmetry for the parabola can be found using the formula:
$$
\text{Line of Symmetry} = x = -$\frac{b}{2a}$.
$$
For the function $y = x^2$, where $a = 1$ and $b = 0$, we have:
$$
egin{align*}
\text{Line of Symmetry} &= x = -$\frac{0}{2 \cdot 1}$ \\
$ &= 0.$
$\end{align*}$
$$
Thus, the line of symmetry is also $x = 0$, confirming our earlier observation.
Worked Example: Finding the Turning Point and Line of Symmetry
For the quadratic function $y = 2x^2 - 4x + 1$, we identify:
- $a = 2$, $b = -4$, $c = 1$.
- The line of symmetry is:
$$
egin{align*}
$ x &= -\frac{-4}{2 \cdot 2} \\$
$ &= 1.$
$\end{align*}$
$$
To find the turning point (vertex), we can substitute $x = 1$ back into the function:
$$
egin{align*}
y &= 2(1)^2 - 4(1) + 1 \\
&= 2 - 4 + 1 \\
$ &= -1.$
$\end{align*}$
$$
Thus, the turning point is at (1, -1).
Reading Roots (X-Intercepts) from the Graph
The roots of a quadratic function are the points where the graph intersects the $x$-axis. These roots can be found algebraically by setting the function equal to zero:
$$
ax^2 + bx + c = 0.
$$
To determine the roots you can factor the quadratic, use the quadratic formula, or graphically identify the intersection points with the $x$-axis.
Worked Example: Finding Roots
Consider the quadratic function $y = x^2 - 5x + 6$. To find the roots:
- Factor the equation:
$$
(x - 2)(x - 3) = 0.
$$
- Set each factor to zero:
$$
egin{align*}
$ x - 2 &= 0 $
ightarrow x = 2, \n x - 3 &= 0
$ightarrow x = 3.$
$\end{align*}$
$$
Thus, the roots are $x = 2$ and $x = 3$. Graphing shows these x-intercepts on the $x$-axis at the points (2, 0) and (3, 0).
Conclusion
In this lesson, we have explored the characteristics of quadratic functions and their graphs. We learned how to:
- Visualize and interpret the graph of $y = x^2$.
- Plot a quadratic function from a table of values.
- Identify the turning point and the line of symmetry of a parabola.
- Find the roots of a quadratic function from its graph.
Understanding these concepts will give you a solid foundation for exploring more complex mathematical topics. Quadratic functions form the basis for many applications across different fields.
Study Notes
- Quadratic functions can be expressed in the standard form $f(x) = ax^2 + bx + c$.
- The graph of a quadratic function is called a parabola. The vertex represents the turning point.
- The line of symmetry for a quadratic function is given by $x = -\frac{b}{2a}$.
- Roots or x-intercepts occur where the graph intersects the x-axis and can be found using algebraic methods or graphically.
- The shape of the parabola (opening upwards or downwards) is determined by the sign of the coefficient $a$.