Equations for the gradient of a straight line
gradient = rise/run = y2-y1/x2-x1
Under what conditions does a straight line make an angle of 45* with the horizontal axis?
When the line has magnitude 1 or -1, but only when the coordinate axes have EQUAL scales.
What are the equations of a vertical or horizontal line, and what are the gradients of the lines?
Horizontal line - all points have same y coordinate of 4, so y = 4
Vertical line - all points have same x coordinate of 3, so x=3
the gradient of a vertical line is undefined (as you can’t divide by 0)
horizontal lines have a gradient of 0
How are the gradients of perpendicular lines related?
Gradient of line 1 x Gradient of line 2 = -1
the gradients of any two perpendicular lines have a product of -1
What does the gradient of a straight line graph/linear model give you? What about units?
The rate of change of the quantity on the vertical axis with respect to the quantity on the horizontal axis.
Gradients have units - these are obtained by combining the simpler units of the labels on the x and y axes. These resulting unit is called a derived unit.
What is the quadratic formula?
x = (-(b) ± sqrt(b^2 - 4ac)) / 2ac
Points to obtain to sketch the graph of a quadratic equation.
- Find any X intercept(s)
Set y = to 0 and solve for x. May wish to calculate discriminant to determine how many real roots there are.
Can solve using FACTORISATION, COMPLETING THE SQUARE or the QUADRATIC FORMULA. - y intercept
- the vertex of the parabola
First, find the axis of symmetry (this gives you the x coordinate for the vertex).
Next, plug in this x value to get the y coordinate of vertex
If there is only one root (preventing axis of symmetry from being found), then use completing the square method to find the x coordinate of the vertex, then plug x value into equation to get the y coordinate of the vertex.
In completed square form, the x value of the vertex is the opposite sign of the second number (the non x term) within the brackets.
Rules for using the discriminant to determine how many real roots a quadratic equation has
Discriminant = b^2-4ac
if discriminant > 0, there are multiple real roots
if discriminant = 0, there is one real root (repeated root)
if discriminant < 0, there are no real roots
If you cannot factorise using integers but were expecting roots based on discriminant, don’t worry, you CAN have a situation where there are real roots but the equation is not factorisable with integers - in these cases use the quadratic formula.
What is a vertex?
The highest point on a n-shaped parabola
The lowest point on a u-shaped parabola
Why can quadratic equations be solved by factorising?
If the product of two or more numbers is 0, then at least one of the numbers must be 0.
How to express a quadratic in completed square form.
For ax^2 + bx + c
- First, if a ≠ 1, take out a factor of a. Remember to include the c term within the brackets!
- write (x ± 0.5b)^2 - (0.5b)^2 + c
Inside the bracket write half of the b term coefficient (the sign inside the brackets depends on +ve or -ve destination of coefficient b) - subtract the square of the number written inside the brackets in the previous step, and remember to add the c term back in.
(x ± 0.5b)^2 - (0.5b)^2 + c - collect the constant terms
What is significant about whether a quadratic equation has ‘real’ solutions or not?
If there are no real solutions, then the graph does not touch or intercept the x axis.
If it has one or more real solutions, then the graph intercepts the x axis at one or more points.
