Lesson 2: Understanding Quadratic Functions and Parabolas
Lesson 2: Quadratic Functions and Parabolas
Features of a parabola: vertex, axis of symmetry and intercepts.
What is a Quadratic Function?
A quadratic function is a function of the form f(x) = ax2 + bx + c, where “a” cannot be 0. The cool thing about these functions is that their graphs form a shape called a parabola! Parabolas may open upward or downward. They have the “U” shape.
How to Graph a Parabola
The most basic parabola has an equation f(x) = x2. To plot a parabola, you can start by creating a table of values. Pick various x-values, plug them into the function, and then find the corresponding f(x)-values. Once you have a bunch of points, you can graph them on a coordinate plane and see the beautiful curve take shape!
What’s the Vertex?
The vertex is a significant part of the parabola—it’s the highest or lowest point, depending on the direction it opens. If you’re looking at a quadratic function in vertex form:
f(x) = a(x – h)2 + k Vertex (h, k)
You can easily spot the vertex at the point (h, k). This makes it super easy to graph the function!
Changing Standard Form to Vertex Form
Sometimes, we’ll need to convert a quadratic from its standard form to vertex form to find that helpful vertex. This is done through a process called “completing the square.” It’s like rearranging the equation a little bit—don’t worry; it’s easier than it sounds! You can add and subtract the same number to keep things balanced, which helps you identify the vertex quickly.
Completing the square can also help us solve quadratic equations to find their zeros, or x-intercepts. These are the points where the parabola crosses the x-axis, and they can be vital in figuring out where something (like a score in a game) becomes zero.
Real-Life Applications of Quadratics
You might be surprised at how often quadratic functions pop up in real life! For example, if you’ve ever played the “Angry Birds” game, you know that understanding how objects fly in a parabolic path is key to making it fun and realistic. The same goes for the path of a soccer ball when it’s kicked or a diver making a splash off a diving board. In basketball, players use quadratic functions to figure out the perfect angle and force to make a basket—pretty amazing, right?
Wrapping It Up
Quadratic functions are not just about numbers; they’re a fascinating part of the world around us! In the coming lessons, we’ll dive deeper into how to solve quadratic equations using methods like factoring and the Quadratic Formula. These tools will make it even easier to find those important x-intercepts. As we explore this topic together, keep in mind how these concepts connect to real-life scenarios, from sports to games and much more!
Key ideas: Quadratic Functions and Parabolas
A quadratic relation can be written in three forms, and each one shows different features of its parabola:
- Standard form y = ax2 + bx + c: the y-intercept is c.
- Vertex form y = a(x − h)2 + k: the vertex is (h, k) and the axis of symmetry is x = h.
- Factored form y = a(x − r)(x − s): the zeros (x-intercepts) are r and s, and the axis of symmetry is halfway between them, x = (r + s)/2.
In every form, a tells the direction: a > 0 opens up (the vertex is a minimum); a < 0 opens down (the vertex is a maximum). The larger |a| is, the narrower the parabola.
Worked examples
Example 1
State the key features of y = −2(x − 3)2 + 8.
- Vertex (3, 8); axis of symmetry x = 3; a = −2 < 0, so it opens down and the maximum value is 8.
- Zeros: 0 = −2(x − 3)2 + 8 → (x − 3)2 = 4 → x − 3 = ±2 → x = 1 or x = 5.
- y-intercept: −2(0 − 3)2 + 8 = −10.
Example 2
Complete the square to find the vertex of y = x2 + 6x + 5.
- Half of 6 is 3, and 32 = 9: y = (x2 + 6x + 9) − 9 + 5.
- y = (x + 3)2 − 4.
- Vertex (−3, −4), minimum value −4.
Example 3
Find the vertex of y = (x − 2)(x + 4).
- Zeros: x = 2 and x = −4.
- Axis of symmetry: x = (2 + (−4))/2 = −1.
- y = (−1 − 2)(−1 + 4) = (−3)(3) = −9. Vertex (−1, −9)
Example 4
A ball’s height is h = −5t2 + 20t + 1 (metres, seconds). What is its maximum height?
- Complete the square: h = −5(t2 − 4t) + 1 = −5(t − 2)2 + 20 + 1.
- h = −5(t − 2)2 + 21.
- The ball reaches 21 m at t = 2 s.
Common mistakes
- Reading the vertex of y = (x + 3)2 − 4 as (3, −4); the sign inside the bracket flips: the vertex is (−3, −4).
- Forgetting to subtract the number you added when completing the square.
- Thinking a parabola always has two zeros; it can have one or none.
- Mixing up the y-intercept (set x = 0) with the zeros (set y = 0).
Questions students ask
How do I know if the vertex is a maximum or a minimum?
Look at a: if a is negative, the parabola opens down and the vertex is a maximum; if positive, it is a minimum.
Which form is “best”?
It depends on the question: vertex form for the max/min, factored form for the zeros, standard form for the y-intercept.
Grade 10 Parabola Quiz
Check your understanding
Try the quiz for this lesson. You’ll see right away if each answer is correct.