Quadratic Functions Theory (MYP)

Question 1

What is a quadratic?

Answer 1

The general quadratic looks like:

y=ax² + bx + c

“Quad” means “square”, which in math means to the power of 2. So a quadratic is an equation that contains an x², and this is the highest power. There can’t be, for example x³, nor x⁴, and so on. Only the square is the highest power. Quadratic expression contains three terms, which makes it a trinomial.

Question 2

What is a coefficient?

Answer 2

A coefficient is a number that multiplies (it is attached to) a variable, like an x or any other letter.

e.g.

3x² + 2x - 4

3 is the coefficient of x²,

and 2 is the coefficient of x.

Question 3

What is the leading coefficient?

Answer 3

The leading coefficient, is the number that is attached to the x with the highest power.

e.g. in the trinomial

4x² + 3x + 9

the leading coefficinent is 4

Question 4

What is a constant?

Answer 4

A constant is a number that is not attached to a variable (like an x, or any other number).

e.g. 3x² + 4x + 7

7 is the constant.

3 and 4 are coefficients.

3 is the leading coefficient.

Question 5

What does the graph of f(x)=x² look like? What are the vertex and line of symmetry?

Answer 5

Vertex: (0,0)

Line of symmetry: x=0

Question 6

How do you shift f(x)=x² to the right by h?

Answer 6

You put a “-h” in with the x term.

g(x)=(x-h)²

Question 7

How do you find the y-intercept of ANY function?

Answer 7

You plug in 0 for x and solve for y….because when you’re on the y-axis, the x-coordinate MUST be equal to 0.

Question 8

How do you find the x-intercept of ANY function?

Answer 8

You plug in 0 for y and solve for x….because when you’re on the x-axis, the y-coordinate MUST be equal to 0.

We care a lot about the x-intercept of quadratic functions. They even have three different names:

• zero
• root
• x-intercept

Question 9

How do you shift f(x)=x² to the up by k?

Answer 9

You put a “+k” at the end.

g(x)=x²+k

Question 10

How do you mirror f(x)=x² over the x-axis?

Answer 10

You multiply the whole function by -1.

g(x)= -(x²).

Question 11

What are the three ways to solve a quadratic equation (aka find the x-intercepts of the function)?

Answer 11

• Factorization
• Completing the square
• The quadratic formula

Question 12

How do you vertically stretch f(x)=x² by a?

Answer 12

You multiply the whole function by a.

g(x)=a(x²).

Question 13

What are the steps to finding the roots by factorising?

Answer 13

1. Get everything on one side, leaving just 0 on the other. (i.e. x²-2x+1=0).
2. Factorise, so you get something like (x+a)(x+b)=0.
3. Solve each parenthesis for x. (i.e. x+a=0 and x+b=0)

Question 14

What’s the quadratic formula?

Answer 14

Question 15

When do you need to use the “complete the square” method?

Answer 15

Whenever you need the form of the equation to tell you the vertex.

f(x)=(x-h)²+k

Question 16

What are the steps for factorising a quadratic equation?

ax²+bx+c=0

Answer 16

1. Factor out any common factors. (So anything that goes into all of the terms. i.e. 6x²-2x-8 = 2 (3x²-x-4))
2. Make a list of all the factors that multiply to make ac.
3. Check to see which two factors from step two add to the b term.
4. Rewrite the equation, separating the middle term into the two factors you found.
5. Take the common factors out of the first two terms and the last two, and regroup.

For example, 3x²-x-4:

1. 6x²-2x+8=2(3x²-x-4). Now I need to factorise 3x²-x-4
2. 3•-4=-12, so the factors are 1•-12, -1•12, 2•-6, -2•6, 3•-4, -3•4
3. Do any of the pairs add to -1? Yep, 3+-4=-1.
4. 3x²+3x-4x-4
5. 3x(x+1) - 4(x+1) = (3x-4)(x+1). So the answer is 6x²-2x+8=2(3x-4)(x+1).

Question 17

How do you know when there are two real roots for y=ax²+bx+c?

(aka, the graph crosses the x-axis two times)

Answer 17

You find when the discriminant greater than 0.

So solve b²-4ac>0.

Question 18

How do you know when there are two repeated real roots for y=ax²+bx+c?

(aka, the graph touches the x-axis just once)

Answer 18

You find when the discriminant is 0.

So solve b²-4ac=0.

Question 19

How do you know when there are no real roots for y=ax²+bx+c?

(aka, the graph never crosses the x-axis)

Answer 19

You find when the discriminant less than 0.

So solve b²-4ac<0.

Question 20

How do you turn

f(x)=ax²+bx+c

into the “vertex” form

f(x)=a(x-h)²+k?

Answer 20

Question 21

What are the three forms a quadratic function can take?

Remember, these are all the same function, it’s just wearing different uniforms.

Answer 21

• Standard form: y=ax²+bx+c
• Vertex form: y=a(x-h)²+k
• Intercept form: y=a(x-p)(x-q)

Question 22

What does the standard form of a quadratic function tell us right away about the graph?

y=ax²+bx+c

Answer 22

Right away we know that the y-intercept is at c.

Question 23

How can we use the vertex form to graph a quadratic?

y=a(x-h)²+k

Answer 23

Question 24

How can we use the intercept form to graph a quadratic?

y=a(x-p)(x-q)

Answer 24

Question 25

How can we use the standard form to graph a quadratic?

y=ax²+bx+c

Answer 25