Transformation
If the basic parabola
f(x) = x^2
is translated 3 units to the right and 2 units down, write down the equation of the newly translated graph.
y=(x-3)^2-2
Solve the linear inequation
7 - 3x \ge 16
x<=-3
Given the function f(x) = 3x^2 - 2x + 1 , evaluate f(-2).
f(-2)=17
Express the inequality -3 < x \le 7 using interval notation.
(-3, 7]
Expand and simplify 4(3x - 2) - 2(x + 5).
10x-18
Determine algebraically whether the function g(x) = 4x^3 - 2x is an even function, an odd function, or neither. You must show your working.
Odd
Solve the quadratic equation x^2 - 2x - 15 = 0 by factorising.
x = 5 or x = -3
Find the equation of the straight line that passes through the points (1, 4) and (3, 10). Give your answer in the form y = mx + c.
y=3x+1
Simplify the expression \sqrt{12} + \sqrt{27} into a single surd.
5sqrt(3)
Factorise completely 3x^2 - 12 .
3(x-2)(x+2)
Given the linear function h(x) = 3x - 6 , find the coordinates of the x-intercept and y-intercept for the graph of the absolute value function y = |h(x)| .
x-int (2,0)
y-int (0,6)
Determine the values of the constant k for which the quadratic equation
x^2 + kx + 16 = 0
has exactly one real (repeated) root.
k=8 or -8
By completing the square, express the quadratic function y = x^2 + 6x + 5 in the turning point form y = (x + h)^2 + k . Hence, state the coordinates of its turning point.
y=(x+3)^2-4
Vertex(-3,-4)
Expand and simplify the expression (3\sqrt{2} - \sqrt{5})(3\sqrt{2} + \sqrt{5}) .
13
Simplify fully \frac{x^2 - x - 12}{x^2 - 16} .
(x+3)/(x+4)
Let f(x) = \sqrt{x} and g(x) = x^2 - 16 . Find the equation for the composite function f(g(x)) and state its natural domain.
f(g(x))=sqrt(x^2-16)
x<=-4 or x>=4
Solve the quadratic inequation x^2 + x - 12 < 0 and state the solution algebraically.
-4<x<3
Consider the rational function y = \frac{2}{x - 3} + 1 . State the equations of the vertical and horizontal asymptotes, and clearly define the domain and range of this function.
Vertical Asymptote: x = 3.
Horizontal Asymptote: y = 1.
Domain: x \ne 3.
Range: y \ne 1 .
Rationalise the denominator and fully simplify \frac{10}{\sqrt{5}} .
2sqrt(5)
Solve the quadratic equation 2x^2 - 7x - 15 = 0 .
x = 5 or x = -\frac{3}{2}
The graph of an unknown function y = f(x) undergoes the following three transformations, strictly in the order given below:
A reflection in the x-axis.
A horizontal dilation by a factor of \frac{1}{2} .
A translation 4 units to the left.
Write the equation of the final transformed graph in terms of f(x).
y = -f(2(x + 4))
Find the values of the constants A, B, and C if the following identity holds true for all values of x: 2x^2 - 5x + 7 \equiv A(x-2)^2 + B(x-2) + C
A=2, B=3, C=5
A circle has the expanded equation x^2 + y^2 - 4x + 6y - 12 = 0 . By rearranging this into the center-radius form, find the coordinates of its center and the length of its radius. Then, mathematically determine if the point (5, 1) lies inside, outside, or on the circle.
Center (2, -3), Radius 5. The point lies on the circle.
Rationalise the denominator and fully simplify \frac{6}{3 - \sqrt{3}} .
3+sqrt(3)
Solve the simultaneous equations y = x^2 - x - 5 and y = x + 3 .
x=-2, y=1 or
x=4, y=7