Mathematics • Year 9 • Unit 2 • Lesson 15
Comparing Non-Linear Graphs
Build the four-family fingerprint habit: parabola, circle, hyperbola, exponential. Watch one worked example identifying four equations, fill in a guided one matching a sketch, then run eight independent classify-and-justify problems.
1. I do, fully worked example
Read every line. The trick is to scan for the SIGNATURE term ($x^2$, sum of squares, $k/x$, or $a^x$) before doing any plotting.
Problem. Identify the family of each equation and state ONE distinguishing key feature: (a) $y = (x - 3)^2 - 4$ (b) $x^2 + y^2 = 25$ (c) $y = 8/x$ (d) $y = 2^x$.
Step 1, Equation (a): $y = (x - 3)^2 - 4$.
Bracket squared $\Rightarrow$ parabola (vertex form). Key feature: vertex at $(3, -4)$.
Reason: $x^2$ pattern means a U-shape.
Step 2, Equation (b): $x^2 + y^2 = 25$.
Both $x^2$ and $y^2$ summed, positive right side $\Rightarrow$ circle. Key feature: centre $(0, 0)$, radius $5$.
Reason: sum of squares is the Pythagorean signature.
Step 3, Equation (c): $y = 8/x$.
$x$ in the denominator $\Rightarrow$ hyperbola. Key feature: asymptotes $x = 0$, $y = 0$; branches in Q1 and Q3 ($k = 8 > 0$).
Reason: inverse-variation form gives two branches.
Step 4, Equation (d): $y = 2^x$.
$x$ as the exponent $\Rightarrow$ exponential. Key feature: $y$-intercept $(0, 1)$; asymptote $y = 0$.
Answer: (a) parabola, (b) circle, (c) hyperbola, (d) exponential.
2. We do, fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank. 4 marks
Problem. Identify the family and give one feature for each of: (a) $y = -2(x + 1)^2 + 3$ (b) $x^2 + y^2 = 100$ (c) $y = -6/x$ (d) $y = 3^x$.
(a) Signature term: __________________ (look for the squared bracket). Family: __________________ . Key feature: vertex $($ ______ $,$ ______ $)$.
(b) Signature: BOTH __________________ and __________________ summed. Family: __________________ . Key feature: centre $(0, 0)$, radius $= \sqrt{100} = $ ______ .
(c) Signature: $x$ in the __________________ . Family: __________________ . $k = -6$ so branches in quadrants __________________ and __________________ .
(d) Signature: $x$ in the __________________ . Family: __________________ . $y$-intercept: $($ ______ $,$ ______ $)$ (because $a^0 = 1$).
3. You do, independent practice
Show your working under each problem. 3.1–3.4 are foundation (single-equation classify). 3.5–3.6 are standard (classify + one feature). 3.7–3.8 are extension (mixed groups and side-by-side comparisons).
Foundation, name the family
3.1 Name the family: $y = -3/x$. 1 mark
3.2 Name the family: $x^2 + y^2 = 49$. 1 mark
3.3 Name the family: $y = 5^x$. State the $y$-intercept. 1 mark
3.4 Name the family: $y = (x + 4)^2 - 1$. State the vertex. 1 mark
Standard, family plus one feature
3.5 For each, name the family and state ONE distinguishing feature: (a) $y = -(x + 2)^2 + 5$ (b) $x^2 + y^2 = 36$ (c) $y = 10/x$. 2 marks
3.6 Which families NEVER have an $x$-intercept? Justify by checking each of parabola, circle, hyperbola and exponential. 2 marks
Extension, comparison tables
3.7 Complete a side-by-side comparison table for $y = x^2$ and $y = 2^x$: state (a) the $y$-intercept of each, (b) the $x$-intercept (or "none") of each, (c) what happens to each as $x \to -\infty$ (one rises, the other approaches an asymptote, which is which?). 2 marks
3.8 A relationship gives $x: 0, 1, 2, 3, 4$ and $y: 1, 3, 9, 27, 81$. (a) Show using first differences that it is non-linear. (b) Identify which family it belongs to. (c) Find a rule of the form $y = a^x$ that fits every point. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2, We do (faded four-pattern classify)
(a) Signature: $(x + 1)^2$ (squared bracket). Family: parabola. Vertex $(\mathbf{-1}, \mathbf{3})$.
(b) Signature: BOTH $x^2$ and $y^2$ summed. Family: circle. Radius $= \sqrt{100} = \mathbf{10}$.
(c) Signature: $x$ in the denominator. Family: hyperbola. $k = -6$ so branches in Q 2 and Q 4.
(d) Signature: $x$ in the exponent. Family: exponential. $y$-intercept $(\mathbf{0}, \mathbf{1})$.
3.1, $y = -3/x$
Hyperbola ($k = -3$, branches in Q2 and Q4).
3.2, $x^2 + y^2 = 49$
Circle, centre $(0, 0)$, radius $7$.
3.3, $y = 5^x$
Exponential. $y$-intercept $(0, 1)$ (since $5^0 = 1$).
3.4, $y = (x + 4)^2 - 1$
Parabola. Vertex $(-4, -1)$.
3.5, Family + feature
(a) Parabola, vertex $(-2, 5)$ (opens DOWN since $a = -1 < 0$).
(b) Circle, centre $(0, 0)$, $r = 6$.
(c) Hyperbola, asymptotes $x = 0$ and $y = 0$; branches in Q1 and Q3 ($k = 10 > 0$).
3.6, No $x$-intercept families
Hyperbola and exponential never have an $x$-intercept.
Parabola CAN have $x$-intercepts (e.g. $y = x^2 - 4$ at $x = \pm 2$), not ruled out.
Circle CAN cross the $x$-axis (at $(\pm r, 0)$), not ruled out.
Hyperbola $y = k/x$: $y = 0$ would need $k/x = 0$, impossible for any non-zero $k$, never on the $x$-axis.
Exponential $y = a^x$ with $a > 0$: $a^x > 0$ for all $x$, so $y$ is never $0$, never on the $x$-axis.
3.7, $y = x^2$ vs $y = 2^x$
(a) $y$-intercepts: $y = x^2$ has $(0, 0)$; $y = 2^x$ has $(0, 1)$.
(b) $x$-intercepts: $y = x^2$ has $(0, 0)$ (one); $y = 2^x$ has NONE.
(c) As $x \to -\infty$: $y = x^2$ RISES to $+\infty$ (the left arm of the parabola goes up); $y = 2^x$ APPROACHES the asymptote $y = 0$ from above. Same input direction, very different output behaviour, family controls everything.
3.8, Identify $y: 1, 3, 9, 27, 81$
(a) First differences: $2, 6, 18, 54$, NOT constant, so non-linear.
(b) Each $y$ value is $3 \times$ the previous (multiplicative ratio of $3$). Family: exponential.
(c) Rule: $\mathbf{y = 3^x}$. Check: at $x = 0, 1, 2, 3, 4$ we get $1, 3, 9, 27, 81$, all match.