Recognising and Interpreting Algebraic and Trigonometric Graphs

Linear Functions

The graph of a linear function is a straight line. It is most commonly expressed in the form $y = mx + c$. In this equation, $m$ represents the gradient (steepness) of the line, and $c$ represents the y-intercept, which is the point where the line crosses the y-axis.

Quadratic Functions

The graph of a quadratic function $y = ax^2 + bx + c$ forms a characteristic curve known as a parabola. Its orientation is determined by the coefficient $a$. If $a > 0$, the curve is $\cup$ shaped (opening upwards), and if $a < 0$, the curve is $\cap$ shaped (opening downwards).

Every quadratic graph has a single turning point and a vertical line of symmetry that passes through this point. The y-intercept is always at the value $c$. The x-intercepts, or roots, are the solutions to the equation $ax^2 + bx + c = 0$.

The turning point can be found by completing the square to find the minimum or maximum value of the function.

Simple Cubic Functions

A basic cubic function of the form $y = ax^3$ has a distinct shape that passes through the origin. These graphs possess rotational symmetry of order 2 about the origin, meaning the graph looks the same if rotated 180 degrees around $(0,0)$.

The Reciprocal Function

The reciprocal function is defined as $y = \frac{1}{x}$ where $x \neq 0$. Because the denominator cannot be zero, the graph never touches the y-axis. Similarly, $y$ can never be zero, so it never touches the x-axis.

If $x$ is positive, $y$ is also positive. If $x$ is negative, $y$ is negative. Consequently, the curve exists only in the first and third quadrants. The graph passes through the coordinates $(1,1)$ and $(-1,-1)$. As $x$ increases, $y$ decreases towards zero, and as $x$ approaches zero, $y$ increases towards infinity.

The Exponential Function

The exponential function is given by $y = k^x$ for positive values of $k$. Because any number to the power of zero is one ($k^0 = 1$), all exponential graphs pass through the point $(0,1)$.

When $k > 1$, the function represents exponential growth. As $x$ increases, $y$ increases rapidly. For negative values of $x$, the value of $y$ stays between 0 and 1, getting closer to zero as $x$ becomes more negative.

When $0 < k < 1$, the function represents exponential decay. As $x$ increases, $y$ decreases rapidly towards zero. For negative values of $x$, the value of $y$ increases as $x$ becomes more negative.

Trigonometric Functions

Trigonometric functions are periodic, meaning they repeat their shapes at regular intervals. When using degrees:

1. $\sin x$ and $\cos x$: Both vary between a maximum of $+1$ and a minimum of $-1$. Key values include $\sin 0^{\circ} = 0$, $\sin 90^{\circ} = 1$, $\cos 0^{\circ} = 1$, and $\cos 90^{\circ} = 0$.

1. $\tan x$: This function varies from $-\infty$ to $+\infty$. It has a value of 0 at $0^{\circ}$ and approaches infinity as it nears $90^{\circ}$.

Worked Example: Comparing $x^2$ and $x^3$

Problem: Sketch the graphs of $y = x^2$ and $y = x^3$ for the range $0 \leq x \leq 2$.

Solution: Both graphs start at $(0,0)$ and pass through $(1,1)$. For values of $x$ between 0 and 1, such as $x = 0.5$, we find that $x^3 < x^2$ (since $0.125 < 0.25$). For values where $x > 1$, the cubic grows much faster, meaning $x^3 > x^2$.

Worked Example: Intersections

Problem: Determine how many points of intersection exist between $y = \frac{1}{x}$ and $y = 2^x$.

Solution: By sketching both curves on the same axes, we observe their behaviour. The reciprocal curve $y = \frac{1}{x}$ stays in the first quadrant for $x > 0$. The exponential growth curve $y = 2^x$ also stays in the first quadrant. They cross at exactly one point.

Worked Example: Trigonometric Inequalities

Problem: Find the range of values for which $\sin \theta < \cos \theta$ within the interval $0 \leq \theta \leq 180^{\circ}$.

Solution: Sketch both $\sin \theta$ and $\cos \theta$. At $\theta = 0^{\circ}$, $\sin \theta = 0$ and $\cos \theta = 1$, so the condition is met. The graphs intersect when $\sin \theta = \cos \theta$, which occurs at $\theta = 45^{\circ}$. Beyond this point, $\sin \theta$ becomes larger than $\cos \theta$ (which eventually becomes negative). Therefore, the range is $0 \leq \theta < 45^{\circ}$.