Revision and Warm-up
Exponents
In \(a^n\), \(n\) and \(a\) are called the exponent and base respectively.
Rules for exponents
\(a^ma^n=a^{m+n}\)
\(\frac{a^m}{a^n}=a^{m-n}\)
\((a^m)^n=a^{mn}\)
\((ab)^n=a^nb^n\)
\(\left(\frac{a}{b}\right)^n=\frac{a^n}{b^n}\)
Algebraic Identities and Algebraic Expressions
Identities Let \(a\) and \(b\) be real numbers. Then we have
\((a+b)^2=a^2+2ab+b^2\)
\((a-b)^2=a^2-2ab+b^2\)
\((a+b)(a-b)=a^2-b^2\)
Solving Linear Equations
Definition A solution to \(F(x)=0\) is a real number \(x_0\) such that \(F(x_0)=0\).
Definition We say that two equations are equivalent if they have the same solution(s).
Solving Quadratic Equations
A quadratic equation (in one unknown) is an equation that can be written in the form \[ax^2+bx+c=0\] where \(a,b\) and \(c\) are constants and \(a\neq0\). To solve this equations, we can use the Factorization Method or the Quadratic Formula.
Quadratic Formula Solutions to Equation \(ax^2+bx+c=0\) are given by \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}.\] \(b^2−4ac\) is called the discriminant
Remainder Theorem and Factor Theorem
Remainder Theorem If a polynomial \(p(x)\) is divided by \(x−c\), where \(c\) is a constant, the remainder is \(p(c)\).
Factor Theorem \((x−c)\) is a factor of a polynomial \(p(x)\) if and only if \(p(c)=0\).
Solving Linear Inequalities
Definition A solution to an inequality \(F(x)<0\) is a real number \(x_0\) such that \(F(x_0)<0\). The definition also applies to other types of inequalities.
Rules for Inequalities Let \(a,b\) and \(c\) be real numbers. Then the following holds.
If \(a<b\), then \(a+c<b+c\).
If \(a<b\) and \(c>0\), then \(ac<bc\).
If \(a<b\) and \(c<0\), then \(ac>bc\). Note: The inequality is reversed.
If \(a<b\) and \(b≤c\), then \(a<c\).
If \(a<b\) and \(a\) and \(b\) have the same sign, then \(\frac{1}{a}>\frac{1}{b}\).
Lines
A line equation in two unknowns \(x\) and \(y\) is an equation that can be written in the form \[ax+by+c=0\] where \(a,b\) and \(c\) are constants with \(a,b\) not both \(0\). More generally, an equation in two unknowns \(x\) and \(y\) is an equation that can be written in the form \[F(x,y)=0\] where \(F\) is a function (from a collection of ordered pairs into \(\mathbb{R}\)).
Definition An ordered pair (of real numbers) is a pair of real numbers \(x_0\), \(y_0\) enclosed inside parenthesis: \((x_0,y_0)\).
Definition A solution to Equation \(F(x,y)=0\) is an ordered pair \((x_0,y_0)\) such that \(F(x_0,y_0)=0\).
Rectangular Coordinate System Given a plane, there is a one-to-one correspondence between points in the plane and ordered pairs of real numbers. The plane described in this way is called the Cartesian plane or the rectangular coordinate plane.
The \(x\)- and \(y\)-axes divide the (rectangular) coordinate plane into 4 regions (called quadrants):
Quadrant I = \(\{(a;b):a>0~~ \text{and}~~ b>0 \}\);
Quadrant II = \(\{(a;b):a<0~~ \text{and}~~ b>0 \}\);
Quadrant III = \(\{(a;b):a<0~~ \text{and}~~ b<0 \}\);
Quadrant IV = \(\{(a;b):a>0~~ \text{and}~~ b<0 \}\);
Lines in the Coordinate Plane Consider the following equation \[\begin{equation} Ax+By+C=0 \tag{0.1} \end{equation}\]
where \(A,B\) and \(C\) are constants with \(A,B\) not both zero. It is not difficult to see that the equation has infinitely many solutions. Each solution \((x_0,y_0)\) represents a point in the (rectangular) coordinate plane. The collection of all solutions (points) form a line, called the graph of Equation (0.1).
Definition For a non-vertical line \(l\), its slope (denoted by \(m\)) is defined to be \[m=\frac{y_2-y_1}{x_2-x_1}\] where \((x_1,y_1)\) and \((x_2,y_2)\) are any two distinct points lying on the line.
Equations for Lines Let \(l\) be a non-vertical line in the coordinate plane.
Suppose \((x_1,y_1)\) is a point lying on \(l\) and \(m\) is the slope. Then an equation for \(l\) can be written in the form \[y−y_1=m(x−x_1)\] called a point-slope form for \(l\).
Suppose the \(y\)-intercept of \(l\) is \((0,b)\) and the slope of \(l\) is \(m\). Then a point-slope form for \(l\) is \[y−b=m(x−0)\] which can be written as \[y=mx+b\] called the slope-intercept form for \(l\).
Parallel and Perpendicular Lines Let \(l_1\) and \(l_2\) be (non-vertical) lines with slopes \(m_1\) and \(m_2\) respectively. Then
\(l_1\) and \(l_2\) are parallel if and only if \(m_1=m_2\).
\(l_1\) and \(l_2\) are perpendicular to each other if and only if \(m_1\cdot m_2=-1\).
Pythagoras Theorem, Distance Formula and Circles
Pythagoras Theorem Let \(a,b\) and \(c\) be the (lengths of the) sides of a right-angled triangle where \(c\) is the hypotenuse.Then we have \[a^2+b^2=c^2\]
Distance Formula Let \(P=(x1,y1)\) and \(Q=(x2,y2)\). Then the distance \(PQ\) between \(P\) and \(Q\) is \[PQ=(x_2−x_1)^2+(y_2−y_1)^2\]
Equation of Circles Let \(C\) be the circle with center at \((h,k)\) and radius \(r\). Then an equation for \(C\) is \[(x-h)^2+(y-k)^2=r^2\]
Parabola
The graph of \[y=ax^2+bx+c\] where \(a\neq 0\), is a parabola. The parabola intersects the \(x\)-axis at two distinct points if \(b^2−4ac>0\). It touches the \(x\)-axis (one intersection point only) if \(b^2−4ac=0\) and does not intersect the \(x\)-axis if \(b^2−4ac<0\).
If \(a>0\), the parabola opens upward and there is a lowest point (called the vertex of the parabola).
If \(a<0\), the parabola opens downward and there is a highest point (vertex).
The vertical line that passes through the vertex is called the axis of symmetry because the parabola is symmetric about this line.
Systems of Equations
A system of two equations in two unknowns \(x\) and \(y\) can be written as \[ \begin{cases} F_1(x,y)=0\\ F_2(x,y)=0 \end{cases} \]
Usually, each equation represents a curve in the coordinate plane. Solving the system means to find all ordered pairs \((x_0,y_0)\) such that \(F_1(x_0,y_0)=0\) and \(F_2(x_0,y_0)=0\), that is, to find all points \((x_0,y_0)\) that lies on the intersection of the two curves.
To solve a system of two linear equations (with two unknowns \(x\) and \(y\)) \[ \begin{cases} ax+by+c=0\\ dx+ey+f=0 \end{cases} \]
we can use elimination or substitution.