Review of Coordinate Geometry and Non-Linear Relationships

Review of Coordinate Geometry

• The chapter starts with a review of coordinate geometry, building upon concepts studied in Year 9.

• Key techniques to remember include how to determine length, gradient, and midpoint of a line segment are important and was covered in exercise 1.08

Length, Gradient, and Midpoint

• Length of AB (Distance between A and B): $AB = \sqrt{(x<em>2 - x</em>1)^2 + (y<em>2 - y</em>1)^2}$

• Gradient (Slope) of AB: $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$

• Midpoint of AB: $M = (\frac{x<em>1 + x</em>2}{2}, \frac{y<em>1 + y</em>2}{2})$

Graphing Straight Lines

• To graph a straight line, find coordinates of two or more points on the line using its equation.

• The x-intercept is found by setting $y = 0$ and solving for $x$.

• The y-intercept is found by setting $x = 0$ and solving for $y$.

Equations of Lines

• Lines can be uniquely determined by certain conditions, leading to different forms of equations:

  • Gradient-Intercept Form: $y = mx + b$, where $m$ is the gradient and $b$ is the y-intercept.

  • Point-Gradient Form: $y - y<em>1 = m(x - x</em>1)$, where $m$ is the gradient and $(x<em>1, y</em>1)$ is a point on the line.

  • Two-Point Form: $y - y<em>1 = \frac{y</em>2 - y<em>1}{x</em>2 - x<em>1}(x - x</em>1)$, where $(x<em>1, y</em>1)$ and $(x<em>2, y</em>2)$ are points on the line.

  • General Form: $ax + by + c = 0$, where $a$, $b$, and $c$ are integers and a > 0. Only one way to write the equation of a line in general form

Parallel and Perpendicular Lines

• Two straight lines are parallel if their gradients are equal.

• The gradients of two lines are equal if the lines are parallel.

• A condition for two lines to be perpendicular might be that the product of their gradients is equal to $-1$.

• If two lines are perpendicular, the product of their gradients is $-1$. That is, $m<em>1 m</em>2 = -1$ (where neither gradient is zero).

• If the product of the gradients of two lines is $-1$, then the lines are perpendicular.

• Two lines with gradients of $m<em>1$ and $m</em>2$ are:

  • Parallel if $m<em>1 = m</em>2$.

  • Perpendicular if $m<em>1 m</em>2 = -1$ that is $m<em>1 = -\frac{1}{m</em>2}$ where neither $m<em>1$ nor $m</em>2$ can equal zero.

• The symbol for parallel lines is $\parallel$.

• The symbol for perpendicular lines is $\perp$.

Using Coordinate Geometry

• Coordinate geometry techniques can be applied to solve geometric problems.

Special Forms of the Parabola

Parabolas of the form $y = ax^2$

• What is the effect on the graph of varying the value of $a$?.

Parabolas of the form $y = ax^2 + k$

• For the equation $y = x^2 + c$, what is the effect on the graph of varying the value of $c$?

Parabolas of the form $y = (x + a)^2$ and $y = (x + a)^2 + k$

Parabolas of the form $y = ax^2 + bx + c$

Finding the y-intercept

• To find the y-intercept of $y = x^2 + x - 12$, we let $x = 0$.

Finding the x-intercepts

• To find the x-intercepts of $y = x^2 + x - 12$, we let $y = 0$.

• The formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ could have been used to find the x-intercepts.

Finding the axis of symmetry

• The axis of symmetry of the parabola $y = ax^2 +bx+ c$ is given by the equation: $x=-\frac{b}{2a}$.

Finding the vertex (or turning point)

• As the vertex lies on the axis of symmetry, its x-coordinate will be the same as that of the axis of symmetry.

• The y-coordinate can be found by substituting this x-value into the equation of the parabola.

• The vertex can also be found by finding the midpoint of the interval joining the x-intercepts.

• The method of completing the square can also be used to find the vertex.

Exponential Graphs

• Exponential equations involve calculating varying powers of a fixed positive number, often called the base.

• Exponential graphs can be drawn by producing a table of values and plotting the resulting points on a number plane.

• Analyzing the equation of the curve will enable you to predict the shape of the curve.

The Hyperbola

Graphs of the form $y = \frac{k}{x}$

• No value for $y$ exists when $x = 0$. This is because no number can be divided by zero.

• The graph has two parts and the curve approaches two lines (the x- and y-axes) but will never touch them. These lines are called asymptotes.

Graphs of the form $y = \frac{k}{x} + c$

• Moving the curve vertically will also change the horizontal asymptote.

• The curve $y = \frac{k}{x} + c$ is obtained by translating the curve $y = \frac{k}{x}$

  • $c$ units up if $c$ is positive

  • $c$ units down if $c$ is negative.

• The asymptotes are $x = 0$ and $y = c$.

Graphs of the form $y = \frac{k}{x - b}$

• The hyperbola $y = \frac{k}{x - b}$ is obtained by translating the curve $y = \frac{k}{x}$

  • $b$ units to the right if $b$ is positive

  • $b$ units to the left if $b$ is negative.

• The asymptotes are $x =b$ and $y = 0$.

• Notice that when the curve is moved horizontally, the vertical asymptote moves as well.

Circles and Their Equations

• Every point on a circle is the same distance away from the centre.

• The equation of a circle that has centre $(p, q)$ and radius $r$ is given by the equation: $(x-p)^2 +(y-q)^2 =r^2$.

Curves of the Form $y = ax^3$ and $y = ax^3 + d$

• Curves of the form $y = ax^3 + d$ are called cubics because of the $x^3$ term.

Curves of the Form $y = ax^n$ and $y = ax^n + d$

• Graphing curves of the form $y = ax^n$ is simple if you recognise the relationship between the signs of $x$ and $y$ for different values of $a$ and $n$.

• To find the sign of $y$ when $y = ax^n$ we need to realise that:

  • If $n$ is even: $x^n$ is always positive (except for $x = 0$).

  • If $n$ is odd: $x^n$ is positive when $x$ is positive. $x^n$ is negative when $x$ is negative.

• The signs of $a$ and $x^n$ will then determine the sign of $y$.

• As $x$ becomes smaller, $ax^n$ (and therefore $y$) becomes smaller.

• As $x$ becomes larger, $ax^n$ (and therefore $y$) becomes larger.

• If $x = 0$, $y = 0$; and if $y = 0$, $x = 0$.

• Hence, the curve only crosses the x-axis once, at the point $(0, 0)$.

• Curves with an even value of $n$ will have shapes like parabolas ($y = ax^2$).

• Curves with an odd value of $n$ will have shapes like cubics ($y = ax^3$).

• As $n$ increases, the steepness of the curve increases, e.g. $y = x^4$ will be steeper than $y = x^2$.

• The curve with the higher value of $n$ will be above the other curve for all values of $x$ except from $-1$ to $1$.

• As we have already seen in 7:09 for the curve $y = ax^3$, changing $a$ changes the 'steepness' of the curve. Hence, for the same value of $n$, a larger value of $a$ will result in a steeper curve.

• e. g. $y = 2x^5$ will be steeper than $y = x^5$.

• Curves of the form $y = ax^n + d$ can be obtained by translating the curve $y = ax^n$ up or down.

  • If $d$ is positive, translate it up $d$ units.

  • If $d$ is negative, translate it down $d$ units.

Curves of the Form $y = a(x - r)^n$

• If the curve $y = ax^n$ is moved (horizontally):

  • $r$ units to the right, the equation of the new curve is $y = a(x - r)^n$.

  • $r$ units to the left, the equation of the new curve is $y = a(x + r)^n$.

Curves of the Form $y = (x - r)(x - s)(x - t)$

• For all the other values of $x$, $y$ is either positive or negative.

• The three x-intercepts divide the x-axis into four sections.

• In each of these sections the $y$ values will be either positive or negative.

• Testing a point in each of the sections will determine whether that section is positive or negative.

The Intersection of Graphs

• Both graphical and algebraic methods can be used to find the point or points of intersection of a line with a parabola, circle or hyperbola.

Algebraic method

• This method is based on the fact that at a point of intersection the x-coordinates on both curves are equal and the y-coordinates on both curves are equal.

For all the graphs met in this chapter it is important that you can:

• Identify different types of graphs from their equations

• Determine a possible equation of a given graph.

Key features to identify from the equation

• Type of Graph

• Straight line

  • x- and y-intercepts

  • slope

  • horizontal and vertical lines

  • Parabola

  • x- and y-intercepts (when they exist)

  • vertex

  • axis of symmetry

  • concavity

  • Hyperbola

  • x- and y-intercepts (where they exist)

  • asymptotes

  • orientation (Quadrants 1 and 3 or Quadrants 2 and 4)

  • Circle

  • centre

  • radius

  • Cubic

  • basic shape (Is it an increasing or decreasing curve?)

  • Exponential

  • x- and y-intercepts (where they exist)

  • concavity

  • Is it increasing or decreasing?

  • asymptote

Translation: Terms in an equation indicate that the curve has been formed by translating a simpler curve in either a vertical or horizontal direction.

• $y = ax^3 + d$ has been formed by translating $y = ax^3$, $d$ units in a vertical direction.

• $y = a(x - r)^2$ has been formed by translating $y = x^2$, $r$ units in a horizontal direction.

Symmetry: Graphs that contain an even power of $x$ have an axis of symmetry.

• If the $y$-axis is an axis of symmetry then the equation will be identical when $x$ is replaced by $-x$.

• e.g. $y = x^2 + 4$, $y = 1 + x^6$, $x^2 + y^2 = 1$