Full Grade 9 test

A tangent meets a circle at one point and is perpendicular to the radius at that point. A secant intersects the circle at two points.

Triangles AOB and COD have equal opposite angles at O; the parallel bases give another equal angle pair. Match corresponding sides and form a proportion using the bases and the diagonal segments.

Tangent of the elevation angle = height / horizontal distance. Thus height = distance × tangent, and distance = height ÷ tangent. In the shadow problem, the shadow is the horizontal distance.

From two line points find the slope k = (y₂ − y₁)/(x₂ − x₁). Then b = y₁ − kx₁, giving y = kx + b. Check both axis scales when reading a graph.

At x = 0, all terms containing x vanish, so the y-intercept value is the constant term. In a linear cost formula it is the fixed fee; total cost = fixed fee + unit price × quantity. For a quadratic, the intercept is (0, c).

Tangent of the elevation angle = height / horizontal distance. Thus height = distance × tangent, and distance = height ÷ tangent. In the shadow problem, the shadow is the horizontal distance.

Factor the numerator and denominator. Cancel only shared nonzero factors, never individual terms in a sum. Keep the original denominator restrictions even when cancellation changes the denominator.

At the intersection both lines have the same y. Solve k₁x + b₁ = k₂x + b₂, then substitute x into either equation to find y. Check the pair in both equations.

The zeros of a function are the x-values for which y = 0.

When the function is already factorised, use the zero-product rule:

(x − x₁)(x − x₂) = 0.

A product equals zero when at least one factor equals zero. Therefore solve x − x₁ = 0 and x − x₂ = 0 separately.

The denominator product must be nonzero, so neither factor may be zero. From (x − a)(x − b), exclude x = a and x = b.

The vertex form of a parabola is y = a(x − m)² + n.

Its vertex coordinates can be read directly:

V(m, n).

Pay close attention to the sign inside the brackets. For example, if the expression is x − (−3), the x-coordinate of the vertex is −3, not 3.

Factor out x: ax² + bx = x(ax + b) = 0. A product is zero when at least one factor is zero. Thus x = 0 or ax + b = 0, giving x = −b/a.

Each next term in a geometric sequence is the preceding term multiplied by the constant ratio q. Find q by dividing consecutive terms. The nth term is aₙ = a₁ × qⁿ⁻¹.

In a right triangle, cos α = adjacent leg / hypotenuse. The hypotenuse is opposite the right angle; the adjacent leg touches the selected acute angle.

An arithmetic sequence has a constant difference d between consecutive terms. Next term = previous term + d. The nth term is aₙ = a₁ + (n − 1)d because reaching term n from term 1 takes n − 1 steps. d can be negative.

Find the full tree shadow length, rather than using only the difference. Heights and shadows form similar right triangles, so the height ratio equals the shadow length ratio.

An arithmetic sequence has a constant difference d between consecutive terms. Next term = previous term + d. The nth term is aₙ = a₁ + (n − 1)d because reaching term n from term 1 takes n − 1 steps. d can be negative.

To expand the brackets, multiply each term in the first pair of brackets by each term in the second:

(x − a)(x − b) = x² − bx − ax + ab.

Combine the like terms:

(x − a)(x − b) = x² − (a + b)x + ab.

1. Multiply x · x.
2. Combine the two terms containing x: −bx − ax.
3. Multiply the constant terms: (−a)(−b) = ab.

Pay close attention to the signs. If a negative number appears in brackets, for example x − (−3), the expression is equal to x + 3.

For ax² + bx + c = 0, the discriminant is D = b² − 4ac. D > 0 gives two real roots, D = 0 one and D < 0 none. For D ≥ 0, x = (−b ± √D)/(2a). In the given equation the x² coefficient is 1.

Factor the numerator and denominator. Cancel only shared nonzero factors, never individual terms in a sum. Keep the original denominator restrictions even when cancellation changes the denominator.