T1T2 Review — Non-Linear Relationships — Grade 10

For each equation, state whether it is a parabola, hyperbola, circle, or exponential function, then identify the named feature.

(a) Find y when x = 3, x = −5, and x = 1/2.
(b) Find x when y = 6, y = −3, and y = 30.
(c) Do the branches appear in Quadrants 1 & 3 or Quadrants 2 & 4? Explain using the value of k.
(d) State the domain and range.

Write the equation of the graph satisfying each description.

Rewrite each equation in standard form. State the centre and radius.

A town has a population of 5000 and grows at 4% per year. The population after t years is P = 5000 × 1.04^t.

(a) Find the population after 1 year.
(b) Find the population after 10 years, to the nearest whole number.
(c) After how many complete years does the population first exceed 6000? (Use trial values.)
(d) The population formula has the form P = A × b^t. State the values of A and b, and explain what each represents in context.

(a) At x = 1, which function gives a larger value?
(b) At x = 4, which function gives a larger value?
(c) For very large values of x (say x = 100), which function gives a much larger value? Explain.
(d) As x → +∞, what happens to the value of y = 20/x? What is the horizontal asymptote? Does y = 2^x have the same asymptote?

Each description refers to a different graph type (parabola, hyperbola, circle, exponential). Identify each type.

(a) The graph passes through (0, 1), never crosses the x-axis, and curves steeply upward for x > 0.
(b) The graph has two branches that approach, but never touch, both coordinate axes.
(c) The graph is a closed curve, centred at a point, and fails the vertical line test.
(d) The graph has a turning point at (3, −2), opens upward, and crosses the x-axis at two points.

A lake is monitored after a pollution event. The fish population starts at 200 and decreases by 15% each year: N = 200 × (0.85)^t.

(a) Find the population after 2 years, to the nearest whole number.
(b) Find the population after 5 years, to the nearest whole number.
(c) After how many complete years does the population first fall below 100?
(d) The model predicts N approaches 0 but never reaches it. What does this mean in the real-world context, and at what point does the model become unrealistic?

Consider the circle x² + y² = 10 and the hyperbola y = 3/x.

(a) Substitute y = 3/x into x² + y² = 10 and show that the result simplifies to x⁴ − 10x² + 9 = 0.
(b) Factorise x⁴ − 10x² + 9 as (x² − 1)(x² − 9), and hence find all four values of x.
(c) Find the corresponding y-value for each x, and list all four intersection points.
(d) Verify that the point (3, 1) lies on both curves.

Topic Review — Non-Linear Relationships