Grade 9 MATH Term 2 Week 6: Introduction to Quadratic Function | MATATAG 2nd Term or 3rd Quarter

Summary

The lesson introduces quadratic functions, shows how they can model real-life situations, and explains how to graph them using a table of values.

Linear and quadratic relationships

The opening activity compares tables of values. Patterns A and C are linear: their outputs change by a constant amount, and their graphs are straight lines.

Pattern B, with values such as (1, 4, 9, 16), does not have a constant first difference. It follows the rule (y=x^2), so it is quadratic.

A quadratic function has the general form

[ f(x)=ax^2+bx+c,\qquad a\ne 0. ]

Its highest power of (x) is 2. The graph of a quadratic function is a smooth, U-shaped curve called a parabola. It opens upward or downward.

Modeling real-life situations

The instructor presents these general steps for modeling:

1. Define a variable for an unknown quantity.
2. Write an expression for each relevant quantity.
3. Use the situation’s conditions to form an equation or function.
4. Simplify the expression and write the resulting function.

Example: Rectangle area

Let the width be (x) metres. The length is 4 metres greater, so it is (x+4). Since area is length multiplied by width,

[ A(x)=x(x+4)=x^2+4x. ]

Example: Product of two numbers

Let one number be (x), and the other be (n). Their sum is 18, so (x+n=18), which gives (n=18-x). Their product is

[ P(x)=x(18-x)=-x^2+18x. ]

Recognizing and graphing quadratic functions

The lesson asks learners to classify graphs as linear, quadratic, or neither. A quadratic graph is a vertical parabola opening upward or downward; a straight line is linear. Other shapes, including sideways curves, are neither of these forms as presented in the lesson.

The quadratic forms highlighted are:

• (y=ax^2+bx+c), the general form
• (y=ax^2)
• (y=ax^2+b)
• (y=a(x-b)^2)

To graph a quadratic function using a table of values:

1. Choose several (x)-values.
2. Substitute each value into the equation to calculate (y).
3. Record the ordered pairs ((x,y)).
4. Plot the points on a coordinate plane.
5. Connect them with a smooth curve, not straight line segments, and add arrows at the ends.

The lesson demonstrates the method with these functions:

• (y=x^2), which opens upward and has its vertex at ((0,0)).
• (y=2x^2), which also opens upward and has its vertex at ((0,0)).
• (y=x^2+2), which opens upward and has its vertex at ((0,2)).
• (y=-2x^2+8), which opens downward and has its vertex at ((0,8)).
• (y=(x-2)^2), which opens upward and has its vertex at ((2,0)).
• (y=-2(x-4)^2), which opens downward and has its vertex at ((4,0)). Its plotted points include ((2,-8)), ((3,-2)), ((4,0)), ((5,-2)), and ((6,-8)).

Practice

The instructor assigns exercises involving modeling situations with quadratic functions and graphing functions from tables of values. Learners are encouraged to check their work with their teachers.

Speakers and sources featured

• Unnamed instructor/presenter, who teaches the lesson and identifies the channel as Math Easy in the subtitles.
• No guest speakers are featured. The channel name differs from the metadata, which lists Math Isip.