L08 — Mixed Non-Linear Graphs
Key Terms
- Parabola — graph of y = ax² + bx + c; symmetric U-shape with a vertex
- Hyperbola — graph of y = k/x; two separate branches with x = 0 and y = 0 as asymptotes
- Circle — graph of (x−h)² + (y−k)² = r²; not a function (fails vertical line test)
- Exponential function — graph of y = ax (a > 0, a ≠ 1); passes through (0,1) in basic form with y = 0 as asymptote
- Asymptote — a line that the curve approaches but never reaches
- Domain — the set of all possible x-values
- Range — the set of all possible y-values
The Four Non-Linear Graph Types
| Type | General Form | y-intercept | x-intercept(s) | Asymptote(s) | Domain | Range |
|---|---|---|---|---|---|---|
| Parabola | y = ax² + bx + c | (0, c) | 0, 1, or 2 | None | All reals | y ≥ k (if a>0) |
| Hyperbola | y = k/x | None | None | x = 0 and y = 0 | x ≠ 0 | y ≠ 0 |
| Circle | (x−h)²+(y−k)²=r² | If |h| ≤ r | If |k| ≤ r | None | [h−r, h+r] | [k−r, k+r] |
| Exponential | y = ax (a > 0, a ≠ 1) | (0, 1) | None | y = 0 | All reals | (0, ∞) |
Identification Tips
- Parabola: equation has x² but y is linear (not squared). Symmetric U-shape.
- Hyperbola: equation is y = k/x (or xy = k). Two separate branches. Both axes are asymptotes.
- Circle: equation has both x² and y² with the same coefficient and a constant on the right. Not a function.
- Exponential: equation has x as an exponent (in the power). Passes through (0, 1) in basic form. One horizontal asymptote.
Hot Tip: The fastest way to identify a graph type from an equation is to look at WHERE x appears. In a parabola, x is squared and in the base. In a hyperbola, x is in the denominator. In a circle, both x and y are squared. In an exponential, x is in the exponent.
Real-World Models
| Situation | Graph Type | Why |
|---|---|---|
| Path of a projectile | Parabola | Quadratic relationship between height and time |
| Time × workers = constant job time | Hyperbola | Inverse proportion: T = k/W |
| All points at a fixed distance from a centre | Circle | Distance formula gives the standard form |
| Population doubling; radioactive decay | Exponential | Constant percentage increase/decrease per period |
Bringing It All Together
This lesson consolidates the four non-linear graph types studied in this topic: parabolas, hyperbolas, circles, and exponential functions. The core skills are identification (recognising which type an equation represents), feature extraction (finding key values like intercepts, asymptotes, centre, radius), and modelling (choosing the right type for a real-world context).
Worked Example 1: Identify the graph type and state one key feature
For each equation, name the graph type and state one key feature.
(a) y = 4/x (b) x² + y² = 36 (c) y = 5x (d) y = 2(x−3)² + 1
Solution:
(a) Hyperbola — asymptotes are x = 0 and y = 0
(b) Circle — centre (0, 0), radius 6. Not a function.
(c) Exponential — y-intercept (0, 1), horizontal asymptote y = 0
(d) Parabola — vertex at (3, 1), opens upward
For each equation, name the graph type and state one key feature.
(a) y = 4/x (b) x² + y² = 36 (c) y = 5x (d) y = 2(x−3)² + 1
Solution:
(a) Hyperbola — asymptotes are x = 0 and y = 0
(b) Circle — centre (0, 0), radius 6. Not a function.
(c) Exponential — y-intercept (0, 1), horizontal asymptote y = 0
(d) Parabola — vertex at (3, 1), opens upward
Comparing Features
When comparing graph types, pay particular attention to:
- Intercepts: parabolas can have 0, 1, or 2 x-intercepts; hyperbolas and exponentials have no x-intercept; circles may have 0, 1, or 2 x-intercepts depending on their position.
- Functions vs relations: parabolas, hyperbolas, and exponentials are functions; circles are not (they fail the vertical line test).
- Asymptotes: hyperbolas have two (x = 0 and y = 0 in basic form); exponentials have one horizontal asymptote; parabolas and circles have none.
Worked Example 2: Find the equation from a description
Find the equation of:
(a) A hyperbola y = k/x passing through (3, 4)
(b) A circle centred at the origin passing through (5, 12)
Solution:
(a) Substitute (3, 4): 4 = k/3 ⇒ k = 12. Equation: y = 12/x
(b) r² = 5² + 12² = 25 + 144 = 169 ⇒ r = 13. Equation: x² + y² = 169
Find the equation of:
(a) A hyperbola y = k/x passing through (3, 4)
(b) A circle centred at the origin passing through (5, 12)
Solution:
(a) Substitute (3, 4): 4 = k/3 ⇒ k = 12. Equation: y = 12/x
(b) r² = 5² + 12² = 25 + 144 = 169 ⇒ r = 13. Equation: x² + y² = 169
Choosing the Right Model
In real-world problems, the model type depends on the nature of the relationship:
- Parabola: when a quantity has a single maximum or minimum (e.g. height of a thrown ball)
- Hyperbola: when two quantities multiply to a constant (inverse proportion)
- Circle: when all points are a fixed distance from a centre (GPS coverage, radar range)
- Exponential: when a quantity grows or decays by a constant percentage per period (interest, population, medicine clearance)
Worked Example 3: Choose the correct model
A physicist observes that as the distance d from a light source doubles, the intensity I (lux) quarters. Does this relationship follow a parabola, hyperbola, circle, or exponential? Write the equation given I = 100 when d = 1.
Solution:
Intensity follows an inverse-square law: I = k/d². This is still in the hyperbola family (inverse proportion). When d = 1, I = 100 ⇒ k = 100.
Equation: I = 100/d²
Note: "doubles distance ⇒ quarters intensity" confirms inverse proportion (I × d² = constant = 100).
A physicist observes that as the distance d from a light source doubles, the intensity I (lux) quarters. Does this relationship follow a parabola, hyperbola, circle, or exponential? Write the equation given I = 100 when d = 1.
Solution:
Intensity follows an inverse-square law: I = k/d². This is still in the hyperbola family (inverse proportion). When d = 1, I = 100 ⇒ k = 100.
Equation: I = 100/d²
Note: "doubles distance ⇒ quarters intensity" confirms inverse proportion (I × d² = constant = 100).
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Match graphs to equations. Fluency
Four graphs are labelled A, B, C, D below. Match each graph to its equation.
Graph AGraph BGraph CGraph DMatch each graph (A, B, C, D) to its equation:
- (i) y = x² − 4
- (ii) y = 6/x
- (iii) x² + y² = 9
- (iv) y = 2 × (½)x
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Type and key feature. Fluency
For each equation, state the graph type and find the requested feature.
- (a) y = 5/x — state the asymptotes
- (b) y = 3x − 2 — state the horizontal asymptote
- (c) x² + y² = 9 — state the centre and radius
- (d) y = (x − 2)² + 1 — state the vertex
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Domain and range. Fluency
State the domain and range for each function or relation.
- (a) y = 4x
- (b) x² + y² = 25
- (c) y = 3/x
- (d) y = (x + 1)² − 4
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Does it have a y-intercept? Fluency
For each equation, decide whether the graph has a y-intercept. If yes, find it.
- (a) y = 2 × 3x
- (b) y = 5/x
- (c) x² + y² = 16
- (d) y = x² + 3x − 1
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Find the equation from a description. Understanding
Find the equation of the non-linear graph that satisfies the given conditions.
- (a) A hyperbola y = k/x that passes through (2, 6).
- (b) An exponential y = b × ax that passes through (0, 3) and (1, 6).
- (c) A circle centred at the origin that passes through (5, 12).
- (d) A parabola with vertex (2, −3) that passes through (4, 1).
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One point — four graphs. Understanding
Show that the point (2, 4) lies on all four of the following curves by substituting x = 2 into each equation.
- (a) y = x²
- (b) y = 8/x
- (c) y = 2x
- (d) x² + y² = 20
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Choose the correct model. Understanding
For each real-world situation, identify the best graph type (parabola, hyperbola, circle, or exponential) and explain your reasoning.
- (a) The height of a soccer ball over time as it is kicked and lands again.
- (b) The concentration of a medicine in the bloodstream, which reduces by 20% every hour.
- (c) The set of all towns that are exactly 50 km from a central city, shown on a map.
- (d) The time taken to drive a fixed distance as the speed increases (faster speed = less time).
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Which graph type has this property? Understanding
Identify which of the four types (parabola, hyperbola, circle, exponential) matches each description. Each type is used exactly once.
- (a) The graph is not a function — it fails the vertical line test.
- (b) In its basic form y = ax, the graph always passes through (0, 1).
- (c) The graph has exactly two asymptotes, both of which are coordinate axes.
- (d) The graph has a single vertex which is either its maximum or minimum point.
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Gas pressure and volume. Problem Solving
In an experiment, the pressure P (kPa) and volume V (litres) of a fixed amount of gas are recorded:
V (litres) 1 2 4 8 P (kPa) 120 60 30 15 - (a) Calculate PV for each pair of values. What do you notice? What type of relationship is this?
- (b) Write the equation linking P and V.
- (c) Find V when P = 40 kPa.
- (d) A different gas follows the rule P = 120 × (0.5)V. Find P when V = 3. Is this model a hyperbola or an exponential function? What key difference distinguishes the two?
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Satellite dish design. Problem Solving
The cross-section of a satellite dish is parabolic. The dish is 8 m wide and 2 m deep at the edges. Set up coordinates with the vertex of the parabola at the origin and the axis of symmetry along the y-axis (opening upward), so the rim passes through (4, 2) and (−4, 2).
- (a) Use the point (4, 2) to find the equation of the parabola in the form y = ax².
- (b) How deep is the dish (value of y) at a horizontal position of x = 3 m from the centre?
- (c) At what depth y = 1.125 m, how wide is the dish? (Find the two x-values and compute the width.)
- (d) A larger model doubles the width to 16 m (rim at x = 8). Using the same equation, how deep would this larger dish be at its rim?