Ontario 10 Applied (MFM2P)
Lesson

As we have seen, variables can appear in equations with no powers, ($x$x, $y$y, or $z$z for example) in squares and cubes, ($x^2$x2, $y^2$y2, $z^3$z3 for example) or even higher or rational powers ($x^5$x5, $y^{10}$y10, $z^{\frac{1}{2}}$z12).

### Term

The expression $3x+6$3x+6 has $2$2 terms, it has the $3x$3x and the $6$6. We already know from our work in algebra that we can collect, add, subtract, multiply and divide like terms.

A binomial expression has $2$2 terms, while a trinomial expression has $3$3 terms.

### Coefficients and Variables

The variables, or variables, are the letters or unknowns in an expression or equation.

The coefficient is the number (whole, rational, irrational, real) in front of a variable.

### Degree

We call the highest power of an equation the degree of the equation.

Let's look at a few examples:

$y=3x-4$y=3x4 has degree $1$1

$y=x^2+2x+1$y=x2+2x+1 has degree $2$2

$y=-4x^3+3x^2-8x+16$y=4x3+3x28x+16 has degree $3$3

$y=x^4-x^3-x^2+x+7$y=x4x3x2+x+7 has degree $4$4

$y=x^{10}-x^6+x^2+12$y=x10x6+x2+12 has degree $10$10

We would define the degree of a polynomial to be equal to the highest power that is present in the equation.

A quadratic equation is a type of polynomial that has a degree of $2$2.

The following are all examples of quadratics:

$x^2$x2

$5x^2$5x2

$\frac{x^2}{6}+3x-7$x26+3x7

$2x+7-18x^2$2x+718x2

## What Do Quadratics Look Like?

All quadratic equations have common properties, apart from just having a degree of $2$2. Use this interactive to explore different quadratic equations. Can you see the common physical properties?

At this stage, all we really want to explore is the shape of the function.

## What are solutions ?

We can interpret the solution to an equation in $2$2 different ways:

• It is the values of $x$x that makes $y=0$y=0 (an algebraic explanation).
• It is the $x$x-intercepts of the graph (a graphical explanation).

Using the above applet, make the following graphs. See how many $x$x-intercepts each quadratic has.

1. $y=x^2+6$y=x2+6
2. $y=-2\left(x+2\right)\left(x-6\right)$y=2(x+2)(x6)
3. $y=\left(x+4\right)^2$y=(x+4)2

What did you notice?

Hopefully you saw that some quadratics cross the $x$x-axis, some only touch the axis once and some do not cross at all. That is, they can have no, $1$1 or $2$2 $x$x-intercepts. We can think of these intercepts as being the solutions to the quadratic equation.

From only an equation we would have to solve it (set it equal to $0$0 and solve for $x$x) to find out algebraically how many solutions it has, but if we have a graph it is much easier to tell. The next few chapters will help us with the algebraic solutions.

Let's have a look at these worked examples.

##### Question 1

Is $7x^3+3x^2$7x3+3x2 a quadratic expression?

1. Yes

A

No

B

Yes

A

No

B

##### Question 2

Which of the following equations represent a quadratic relationship between $x$x and $y$y?

1. $y=1-x^2$y=1x2

A

$y=\left(x^2-1\right)^3$y=(x21)3

B

$y=x^2+\frac{2}{x}$y=x2+2x

C

$y=\sqrt{x^2+2}$y=x2+2

D

$y=2x^2$y=2x2

E

$y=1-x^2$y=1x2

A

$y=\left(x^2-1\right)^3$y=(x21)3

B

$y=x^2+\frac{2}{x}$y=x2+2x

C

$y=\sqrt{x^2+2}$y=x2+2

D

$y=2x^2$y=2x2

E

##### Question 3

What relationship between $x$x and $y$y is represented by the given graph?

1. Linear

A

B

Other

C

Linear

A

B

Other

C

### Outcomes

#### 10P.QR2.02

Determine, through investigation using technology, that a quadratic relation of the form y = ax^2 + bx + c (a ≠ 0) can be graphically represented as a parabola, and determine that the table of values yields a constant second difference