Mathematics • Year 9 • Unit 2 • Lesson 16
Graphing Non-Linear Relationships
Drill the 5-step graphing recipe: identify family $\to$ choose $x$-values $\to$ build table $\to$ choose scale $\to$ plot and join smoothly. One worked, one guided, eight independent — parabolas, hyperbolas, exponentials.
1. I do — fully worked example
Read every line. Each step shows why we chose those $x$-values and that scale, not just what the answer is.
Problem. Graph $y = x^2 - 4$ using a table of values. Choose a sensible scale and label the intercepts.
Step 1 — Identify the family.
$x^2$ as the highest power, no $x$ in a denominator or exponent. Family: parabola, vertex on the $y$-axis (no $bx$ term), opens up ($a = 1 > 0$).
Reason: parabolas come from $x^2$. The constant $-4$ shifts the vertex down to $(0, -4)$.
Step 2 — Choose $x$-values.
$x = -3, -2, -1, 0, 1, 2, 3$ — symmetric integers around the axis ($x = 0$).
Reason: parabolas need symmetric $x$-values so both arms appear.
Step 3 — Build the table.
$y = 9 - 4, 4 - 4, 1 - 4, 0 - 4, 1 - 4, 4 - 4, 9 - 4 = 5, 0, -3, -4, -3, 0, 5$.
Reason: mirror values across $x = 0$ confirm symmetry; vertex at $(0, -4)$.
Step 4 — Choose the scale.
$x$ range $-3$ to $3$: $1$ unit per square. $y$ range $-4$ to $5$: also $1$ unit per square. Standard grid fits.
Reason: max $|y| = 5$, so $9$ vertical squares cover $-4$ to $5$ comfortably.
Step 5 — Plot and join smoothly.
Plot all $7$ points. Join with a smooth U-shape. Label $y$-intercept $(0, -4)$ and $x$-intercepts $(-2, 0)$, $(2, 0)$.
Reason: smooth curve only (no straight segments). Labelling the intercepts is required.
Answer: U-shape through $(-3, 5), (-2, 0), (-1, -3), (0, -4), (1, -3), (2, 0), (3, 5)$.
2. We do — fill in the missing steps
Same five-step recipe, but with the working faded. Fill in each blank. 4 marks
Problem. Graph $y = \dfrac{6}{x}$ for $-6 \le x \le 6$, $x \neq 0$.
Step 1 — Identify the family: $x$ sits in the __________________ of a fraction $\Rightarrow$ family is __________________.
Step 2 — Choose $x$-values: use factors of $6$, both signs, SKIP $x = 0$. So $x = \pm 1, \pm 2, \pm \_\_\_, \pm \_\_\_$.
Step 3 — Build the table: $y$-values $= \_\_\_, \_\_\_, \_\_\_, \_\_\_, \_\_\_, \_\_\_, \_\_\_, \_\_\_$ (in order from $x = -6$ to $x = 6$, skipping $x = 0$).
Step 4 — Choose the scale: $y$-range is $-6$ to $6$, so $\_\_\_$ unit(s) per square on both axes.
Step 5 — Plot and join smoothly: connect the $4$ points in Quadrant __ with one smooth branch and the $4$ points in Quadrant __ with another. The two branches must NEVER touch the asymptotes $x = \_\_\_$ and $y = \_\_\_$.
3. You do — independent practice
Show your working in the space under each problem. The first four are foundation (build a table only). The middle two are standard (table + scale choice). The last two are extension (full 5-step recipe).
Foundation — build the table
3.1 Complete the table for $y = x^2 + 1$ at $x = -2, -1, 0, 1, 2$. 1 mark
3.2 Complete the table for $y = \dfrac{8}{x}$ at $x = -8, -4, -2, -1, 1, 2, 4, 8$. 1 mark
3.3 Complete the table for $y = 3^x$ at $x = -1, 0, 1, 2$. 1 mark
3.4 List the best $x$-values to use for graphing $y = \dfrac{12}{x}$ between $-12$ and $12$. (Hint: factors of $12$, both signs, exclude $0$.) 1 mark
Standard — table and scale
3.5 For $y = 4^x$ at $x = -1, 0, 1, 2, 3$: (a) build the table, (b) state the $y$-range, (c) suggest a sensible $y$-scale (units per square) for a standard grid. 2 marks
3.6 For $y = x^2 - 2x$ at $x = -1, 0, 1, 2, 3$: (a) build the table, (b) state the vertex by inspection. 2 marks
Extension — apply the full 5-step recipe
3.7 Apply the full 5-step recipe to $y = \dfrac{4}{x}$ at $x = -4, -2, -1, 1, 2, 4$. Sketch the graph in the margin and label both asymptotes. 3 marks
3.8 A student wants to graph $y = 3^x$ for $x = -3$ to $x = 4$. (a) Build the table (use fractions for negative $x$). (b) The $y$-values range from $\tfrac{1}{27}$ to $81$ — explain why a $y$-scale of "$1$ unit per square" is unsuitable, and suggest a better one. (c) State the equation of the horizontal asymptote. 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (faded $y = \dfrac{6}{x}$)
Step 1: $x$ sits in the denominator; family is hyperbola.
Step 2: $x = \pm 1, \pm 2, \pm \mathbf{3}, \pm \mathbf{6}$ (factors of $6$, skip $0$).
Step 3: $y = \mathbf{-1, -2, -3, -6, 6, 3, 2, 1}$ (in order from $x = -6$ to $x = 6$, skipping $x = 0$).
Step 4: $\mathbf{1}$ unit per square on both axes.
Step 5: Quadrant 3 for the negative branch, Quadrant 1 for the positive branch. Asymptotes $x = \mathbf{0}$ (vertical) and $y = \mathbf{0}$ (horizontal).
3.1 — $y = x^2 + 1$
$x = -2, -1, 0, 1, 2$ gives $y = 5, 2, 1, 2, 5$. Symmetric across $x = 0$ as expected; vertex at $(0, 1)$.
3.2 — $y = \dfrac{8}{x}$
$x = -8, -4, -2, -1, 1, 2, 4, 8$ gives $y = -1, -2, -4, -8, 8, 4, 2, 1$. ($x = 0$ excluded; $y$-values are reciprocal pairs scaled by $8$.)
3.3 — $y = 3^x$
$x = -1, 0, 1, 2$ gives $y = \tfrac{1}{3}, 1, 3, 9$. ($3^0 = 1$ always; $3^{-1} = \tfrac{1}{3}$.)
3.4 — Best $x$-values for $y = \dfrac{12}{x}$
Factors of $12$, both signs, exclude $0$: $x = \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 12$. (You can use any subset — $\pm 1, \pm 2, \pm 3, \pm 6$ is the standard "$8$-value" set.)
3.5 — $y = 4^x$
(a) $x = -1, 0, 1, 2, 3$ gives $y = \tfrac{1}{4}, 1, 4, 16, 64$.
(b) $y$-range: $\tfrac{1}{4}$ to $64$.
(c) With max $y = 64$, "$1$ unit per square" needs $64$ squares vertically — far too many. Use about $10$ units per square (so $7$ squares fits $0$–$70$).
3.6 — $y = x^2 - 2x$
(a) $x = -1, 0, 1, 2, 3$ gives $y = 3, 0, -1, 0, 3$.
(b) Minimum $y = -1$ at $x = 1$. Vertex $(1, -1)$.
3.7 — $y = \dfrac{4}{x}$ (full recipe)
Step 1: hyperbola, $k = 4 > 0 \Rightarrow$ Q1 + Q3.
Step 2 (already given): $x = -4, -2, -1, 1, 2, 4$.
Step 3: $y = -1, -2, -4, 4, 2, 1$.
Step 4: $y$-range $-4$ to $4$, $x$-range $-4$ to $4$. $1$ unit per square fits a standard grid.
Step 5: smooth branch in Q1 through $(1, 4), (2, 2), (4, 1)$; smooth branch in Q3 through $(-1, -4), (-2, -2), (-4, -1)$. Asymptotes: $x = 0$ (vertical) and $y = 0$ (horizontal).
3.8 — $y = 3^x$ with awkward $y$-range
(a) $x = -3, -2, -1, 0, 1, 2, 3, 4$ gives $y = \tfrac{1}{27}, \tfrac{1}{9}, \tfrac{1}{3}, 1, 3, 9, 27, 81$.
(b) "$1$ unit per square" would need $81$ vertical squares — off any normal page. Use about $10$ units per square (about $9$ squares fits $0$–$90$); the tiny fraction values near $y = 0$ will look like they sit on the $x$-axis but they don't quite touch.
(c) Horizontal asymptote: $y = 0$.