Full Grade 9 test

Percentage grade = vertical rise ÷ horizontal distance × 100. A 12% grade means a 12 m rise over 100 m horizontally. It is not a 12° angle.

Rearrange aₙ = a₁ + (n − 1)d to n = (aₙ − a₁)/d + 1. Subtract the first term from the given value, divide by the common difference and add 1.

Percentage grade = vertical rise ÷ horizontal distance × 100. A 12% grade means a 12 m rise over 100 m horizontally. It is not a 12° angle.

tan α = opposite leg / adjacent leg. Tangent relates both legs directly without using the hypotenuse. Use inverse tangent to find the angle.

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.

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.

A central angle intercepting the same arc as an inscribed angle is twice as large:

central angle = 2 · inscribed angle.

Multiply the given inscribed angle by 2. The resulting angle cannot exceed 360°.

The domain is the set of allowed input values x. A function assigns exactly one y to each x in its domain. Different inputs may share an output, but one input cannot have two different outputs.

The discriminant is an expression formed from the coefficients of a quadratic equation ax² + bx + c = 0. It helps determine the number of real solutions.

Formula: D = b² − 4ac.

1. Identify the coefficients a, b and c, paying attention to their signs.
2. Calculate b².
3. Calculate 4ac and subtract it from b².

If D > 0, the equation has two real solutions; if D = 0, it has one; if D < 0, it has no real solutions.

In a right triangle, sin α = opposite leg / hypotenuse. Divide the opposite leg by the hypotenuse for sine; multiply the hypotenuse by sin α to find the opposite leg.

The vertex is where a parabola changes from increasing to decreasing or vice versa. Read its x and y from the graph. A downward-opening parabola has its maximum y at the vertex, representing the ball maximum height and when it occurs.

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.

A central angle and an inscribed angle intercept the same arc. Their measures are related by:

inscribed angle = central angle / 2.

Divide the given central angle by 2. Before calculating, make sure that both angles really intercept the same arc.

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

Substitute the given x into y = kx + b. Multiply k by x first, then add b. For a trend line, this is a prediction rather than a guaranteed observed value.

Find the required trigonometric function row and angle column. Their intersection contains the answer.

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.