What is a quadratic?
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.
What is a coefficient?
A coefficient is a number that multiplies (it is attached to) a variable, like an x or any other letter.
3x2 + 2x - 4
3 is the coefficient of x2,
and 2 is the coefficient of x.
What is the leading coefficient?
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
What is a constant?
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.
What does the graph of f(x)=x2 look like? What are the vertex and line of symmetry?
Line of symmetry: x=0
How do you shift f(x)=x2 to the right by h?
You put a "-h" in with the x term.
How do you find the y-intercept of ANY function?
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.
How do you find the x-intercept of ANY function?
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:
How do you shift f(x)=x2 to the up by k?
You put a "+k" at the end.
How do you mirror f(x)=x2 over the x-axis?
You multiply the whole function by -1.
What are the three ways to solve a quadratic equation (aka find the x-intercepts of the function)?
- Completing the square
- The quadratic formula
How do you vertically stretch f(x)=x2 by a?
You multiply the whole function by a.
What are the steps to finding the roots by factorising?
- Get everything on one side, leaving just 0 on the other. (i.e. x2-2x+1=0).
- Factorise, so you get something like (x+a)(x+b)=0.
- Solve each parenthesis for x. (i.e. x+a=0 and x+b=0)
What's the quadratic formula?
When do you need to use the "complete the square" method?
Whenever you need the form of the equation to tell you the vertex.
What are the steps for factorising a quadratic equation?
- Factor out any common factors. (So anything that goes into all of the terms. i.e. 6x2-2x-8 = 2 (3x2-x-4))
- Make a list of all the factors that multiply to make ac.
- Check to see which two factors from step two add to the b term.
- Rewrite the equation, separating the middle term into the two factors you found.
- Take the common factors out of the first two terms and the last two, and regroup.
For example, 3x2-x-4:
- 6x2-2x+8=2(3x2-x-4). Now I need to factorise 3x2-x-4
- 3•-4=-12, so the factors are 1•-12, -1•12, 2•-6, -2•6, 3•-4, -3•4
- Do any of the pairs add to -1? Yep, 3+-4=-1.
- 3x(x+1) - 4(x+1) = (3x-4)(x+1). So the answer is 6x2-2x+8=2(3x-4)(x+1).
How do you know when there are two real roots for y=ax2+bx+c?
(aka, the graph crosses the x-axis two times)
You find when the discriminant greater than 0.
So solve b2-4ac>0.
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)
You find when the discriminant is 0.
So solve b2-4ac=0.
How do you know when there are no real roots for y=ax2+bx+c?
(aka, the graph never crosses the x-axis)
You find when the discriminant less than 0.
So solve b2-4ac<0.
How do you turn
into the "vertex" form
What are the three forms a quadratic function can take?
Remember, these are all the same function, it's just wearing different uniforms.
Standard form: y=ax2+bx+c
Vertex form: y=a(x-h)2+k
Intercept form: y=a(x-p)(x-q)
What does the standard form of a quadratic function tell us right away about the graph?
Right away we know that the y-intercept is at c.
How can we use the vertex form to graph a quadratic?
How can we use the intercept form to graph a quadratic?
How can we use the standard form to graph a quadratic?