Graphs of functions: linear, quadratic, cubic, reciprocal, exponential

Graphs of Functions: Linear, Quadratic, Cubic, Reciprocal, Exponential 📈

Linear Functions – The Straight‑Line Road 🚗

A linear function has the form $y = mx + c$. The graph is a straight line.

• Slope ($m$) tells how steep the line is. Positive slope → up to the right, negative → down.
• Y‑intercept ($c$) is where the line crosses the y‑axis.

Analogy: Think of a road that goes straight from point A to point B. The slope is how fast you climb or descend.

Quadratic Functions – The Parabolic Roller Coaster 🎢

Quadratic functions are of the form $y = ax^2 + bx + c$. Their graphs are U‑shaped (or upside‑down U if $a<0$).

• Vertex – the highest or lowest point.
• Axis of symmetry – the vertical line that splits the parabola.
• Direction – $a>0$ opens up, $a<0$ opens down.

Analogy: Imagine a bowl of water. The bottom of the bowl is the vertex; the water spreads out symmetrically on both sides.

Cubic Functions – The S‑Shaped Snake 🐍

Cubic functions have the form $y = ax^3 + bx^2 + cx + d$. Their graphs can have two turning points, creating an S‑shaped curve.

• Inflection point – where the curve changes concavity.
• End behaviour – as $x \to \pm\infty$, $y$ follows the sign of $a$.

Analogy: Picture a snake slithering: it bends one way, then the other, creating a smooth S shape.

Reciprocal Functions – The Inverse Mirror 🔁

Reciprocal functions are of the form $y = \frac{k}{x}$. Their graphs consist of two separate curves (branches) in opposite quadrants.

• Vertical asymptote at $x=0$ (the y‑axis).
• Horizontal asymptote at $y=0$ (the x‑axis).
• As $x$ approaches zero, $y$ grows without bound.

Analogy: Think of a mirror that flips the sign of the input: the closer you get to the mirror (x=0), the more extreme the output becomes.

Exponential Functions – The Rocket Launch 🚀

Exponential functions are written as $y = a \, b^x$ where $b>0$ and $b eq 1$. Their graphs rise or fall rapidly.

• Base ($b$) determines the growth rate: $b>1$ grows, $0
• Horizontal asymptote at $y=0$.
• Y‑intercept at $y=a$.

Analogy: Imagine a rocket that doubles its speed every minute – the graph shoots up steeply.

Quick Check: Identify the Function Type

1. Graph passes through points (0,2) and (1,4) and is a straight line. ➜ Linear.
2. Graph has a U‑shape with vertex at (3, -1). ➜ Quadratic.
3. Graph looks like an S‑curve crossing the origin. ➜ Cubic.
4. Graph has two branches, one in the first quadrant and one in the third, approaching both axes asymptotically. ➜ Reciprocal.
5. Graph starts near zero and rises sharply, passing through (0,1) and (2,4). ➜ Exponential.

Great job! 🎉 Keep practicing by sketching each type of graph and noting its key features. Happy graphing!