Summary Grade 10 Mathematics - Financial Math

Comprehensive Grade 10 Mathematics Master Study Guide: Functions and Graphs1. Foundations of Functions and Relations1.1 Fundamental DefinitionsCartesian Plane: Formed by the intersection of a horizontal $x$-axis and a vertical $y$-axis at right angles, dividing the plane into four quadrants (I, II, III, IV).Relation: Any rule or pairing that connects elements of a set of inputs ($x$-values) to elements of a set of outputs ($y$-values), represented as ordered pairs $(x; y)$.Function: A specialized relation in which every valid input value ($x$) corresponds to one and only one unique output value ($y$).The Vertical Line Test: A visual method to determine if a graph represents a function. If any vertical line drawn through the graph intersects it at more than one point, the graph is a relation, not a function.1.2 Notation and TerminologyFunctional Notation: Written as $f(x)$, $g(x)$, or $h(x)$, which reads as "the value of the function $f$ at $x$." This acts interchangeably with the dependent variable $y$.Domain: The complete set of all permissible input values ($x$-values) for which a function is mathematically defined.Range: The complete set of all resulting output values ($y$-values) generated by the function across its domain.Intercepts:$y$-intercept: The point where the graph crosses the vertical $y$-axis, determined algebraically by setting $x = 0$.$x$-intercept: The points where the graph crosses the horizontal $x$-axis (also known as roots, zeros, or solutions), determined algebraically by setting $y = 0$ (or $f(x) = 0$).2. The Linear Function (Straight Line)2.1 Standard Equation FormsGradient-Intercept Form:$$y = mx + c$$(Alternatively written in school curricula as $y = ax + q$)Standard Form:$$Ax + By + C = 0$$2.2 Core Parameters and AnalysisGradient ($m$): Represents the steepness and orientation (direction) of the line. It is calculated using any two distinct points $(x_1; y_1)$ and $(x_2; y_2)$ on the line:$$m = frac{y_2 - y_1}{x_2 - x_1} = frac{text{rise}}{text{run}}$$$m 0$: The line slopes upwards from left to right (increasing function).$m 0$: The line slopes downwards from left to right (decreasing function).$m = 0$: The line is completely horizontal, with the equation $y = c$.Undefined Gradient: A vertical line parallel to the $y$-axis, with the equation $x = k$ (note: this is a relation, not a function).$y$-Intercept ($c$): The exact coordinate point $(0; c)$ where the line intersects the $y$-axis.2.3 Geometric Relationships Between LinesParallel Lines: Two distinct lines never intersect because they share identical gradients:$$m_1 = m_2$$Perpendicular Lines: Two lines intersect at a right angle ($90^{circ}$) if the product of their gradients equals $-1$:$$m_1 times m_2 = -1 quad text{or} quad m_2 = -frac{1}{m_1}$$3. The Quadratic Function (Parabola)3.1 Standard Equation Form$$y = ax^2 + q$$(Where parameter $a neq 0$)3.2 Key Properties and Parameter ImpactsShape and Turning Point:The graph is perfectly symmetrical, sharing an axis of symmetry along the $y$-axis ($x = 0$).The turning point (vertex) lies directly on the $y$-axis at coordinates $(0; q)$.The Effect of Parameter $a$:If $a 0$: The parabola opens upwards resembling a "happy face" (concave up). The turning point represents the absolute minimum value of the function. The range is restricted to $y in [q; infty)$.If $a 0$: The parabola opens downwards resembling a "sad face" (concave down). The turning point represents the absolute maximum value of the function. The range is restricted to $y in (-infty; q]$.Stretch and Compression: The magnitude of $a$ ($vert{}avert{}$) dictates width. Large values of $vert{}avert{}$ stretch the parabola vertically, making it narrow. Values of $vert{}avert{}$ close to $0$ compress it vertically, making it wide.4. The Hyperbolic Function (Hyperbola)4.1 Standard Equation Form$$y = frac{a}{x} + q$$(Where $x neq 0$ and $a neq 0$)4.2 Key Properties and Parameter ImpactsThe Effect of Parameter $a$:If $a 0$: The two symmetrical branches of the hyperbola lie entirely within the 1st and 3rd quadrants (relative to the shifted axes).If $a 0$: The branches lie entirely within the 2nd and 4th quadrants.Asymptotes: Invisible boundary lines that the graph approaches infinitely closely without ever touching or intersecting.Vertical Asymptote: $x = 0$ (the $y$-axis), because division by zero is undefined.Horizontal Asymptote: $y = q$, reflecting the vertical translation of the curve.Domain and Range Restrictions:Domain: $x in mathbb{R}, x neq 0$Range: $y in mathbb{R}, y neq q$Axes of Symmetry: The hyperbola is symmetrical along two diagonal lines passing through the intersection point of its asymptotes, with gradients of $+1$ and $-1$:$$y = x + q quad text{and} quad y = -x + q$$5. The Exponential Function5.1 Standard Equation Form$$y = ab^x + q$$(Where base $b 0$, $b neq 1$, and $a neq 0$)5.2 Key Properties and Parameter ImpactsHorizontal Asymptote: The line $y = q$ acts as the boundary asymptote. As $x$ approaches extreme negative or positive values, the curve flattens out toward this line.The Effect of Base $b$:If $b 1$ and $a 0$: The function represents exponential growth and is increasing from left to right.If $0 b 1$ and $a 0$: The function represents exponential decay and is decreasing from left to right.The Effect of Parameter $a$: Controls the vertical reflection and scaling. If $a 0$, the graph sits strictly above its horizontal asymptote; if $a 0$, it sits strictly below.Domain and Range:Domain: $x in mathbb{R}$ (all real numbers)Range:If $a 0$: $y in (q; infty)$If $a 0$: $y in (-infty; q)$6. Trigonometric Functions6.1 Standard Equation FormsSine Wave: $y = a sin(x) + q$Cosine Wave: $y = a cos(x) + q$Tangent Curve: $y = a tan(x) + q$6.2 Core Wave CharacteristicsAmplitude ($a$): Represents half of the total vertical distance between the maximum and minimum peaks of a wave:$$text{Amplitude} = frac{y_{max} - y_{min}}{2}$$For standard unscaled sine and cosine functions where $a = 1$, the amplitude is $1$.Period: The precise interval along the horizontal $x$-axis required for a periodic graph to complete one full, unbroken cycle of its pattern before repeating.Sine and Cosine functions complete a full wave cycle every $360^{circ}$.Tangent functions complete a full pattern cycle every $180^{circ}$.Tangent Asymptotes: Because $tan(x) = frac{sin(x)}{cos(x)}$, the tangent graph features recurring vertical asymptotes wherever $cos(x) = 0$, specifically located at $x = 90^{circ} + k cdot 180^{circ}$ (for all integers $k$).7. Advanced Analytical Tools: Average Gradient and Distance7.1 Average GradientThe average gradient between any two distinct points $(x_1; y_1)$ and $(x_2; y_2)$ on a nonlinear curve measures the steepness of the secant line connecting them:$$text{Average Gradient} = frac{f(x_2) - f(x_1)}{x_2 - x_1}$$7.2 Vertical Length Between Two GraphsWhen evaluating two functions $f(x)$ and $g(x)$ where $f(x)$ lies above $g(x)$ across a specified interval, the vertical distance length between them at any given $x$-value is calculated as:$$text{Vertical Length} = y_{text{top}} - y_{text{bottom}} = f(x) - g(x)$$8. Comprehensive Step-by-Step Worked ExamplesExample 1: Finding a Linear Equation from Two PointsProblem: Determine the equation of the straight line passing through points $P(-3; -2)$ and $Q(3; 4)$.Solution:Calculate the gradient ($m$) using the gradient formula:$$m = frac{y_2 - y_1}{x_2 - x_1} = frac{4 - (-2)}{3 - (-3)} = frac{4 + 2}{3 + 3} = frac{6}{6} = 1$$Substitute the gradient into the gradient-intercept form:$$y = mx + c implies y = 1x + c implies y = x + c$$Substitute the coordinates of point $Q(3; 4)$ into the equation to solve for $c$:$$4 = (3) + c implies c = 4 - 3 = 1$$Write out the final finalized equation:$$y = x + 1$$Example 2: Comprehensive Parabola AnalysisProblem: Given the quadratic function $f(x) = -2x^2 + 18$:Write down the coordinates of the turning point.Calculate the exact $x$-intercepts.Determine the range and maximum value of $f$.Solution:Turning Point: Since the equation is structured as $y = ax^2 + q$ where $q = 18$ and there is no linear $x$ term, the turning point lies on the $y$-axis at $(0; 18)$.$x$-intercepts: Set $f(x) = 0$:$$-2x^2 + 18 = 0$$Divide the entire equation by $-2$:$$x^2 - 9 = 0 implies (x - 3)(x + 3) = 0$$Therefore, $x = 3$ or $x = -3$. The intercepts are $(-3; 0)$ and $(3; 0)$.Range and Maximum: Because $a = -2 0$, the parabola opens downwards, making the turning point a maximum value. The maximum value is $y = 18$, giving a range of $y in (-infty; 18]$.Example 3: Hyperbola Analysis and Sketching PropertiesProblem: Analyze the hyperbola given by $h(x) = frac{8}{x} - 4$:State the equations of the vertical and horizontal asymptotes.Determine the $x$-intercept of the function.Solution:Asymptotes:Vertical Asymptote: $x = 0$ (derived from the undefined denominator restriction).Horizontal Asymptote: $y = -4$ (derived from the vertical shift parameter $q$).$x$-intercept: Set $h(x) = 0$:$$0 = frac{8}{x} - 4 implies 4 = frac{8}{x}$$Cross-multiply to solve for $x$:$$4x = 8 implies x = 2$$The $x$-intercept coordinate is $(2; 0)$.9. Extended Mastery Practice AssessmentQuestionsLinear Functions: A straight line passes through the point $(2; -5)$ and has a gradient of $m = 3$. Find its complete equation and $y$-intercept.Quadratic Functions: For the parabola given by $g(x) = 3x^2 - 12$, determine:The coordinates of the turning point.The equation of the axis of symmetry.The $x$-intercepts.Hyperbolic Functions: Given $f(x) = frac{-5}{x} + 3$:Write down the domain and range.Identify which quadrants the graph's branches occupy.Exponential Functions: Determine the $y$-intercept and horizontal asymptote of the exponential function $y = 2 cdot 3^x - 6$.Trigonometric Functions: For the wave function $y = -4 cos(x)$:State its amplitude.State its range.Graphical Intersections: Given the functions $f(x) = -x^2 + 9$ and the horizontal line $g(x) = 5$, calculate all points of intersection between the two graphs.Detailed Answers and Solutions to Mastery AssessmentLinear Functions Answer:Start with template: $y = 3x + c$Substitute point $(2; -5)$: $-5 = 3(2) + c implies -5 = 6 + c implies c = -11$.Equation: $y = 3x - 11$ (with a $y$-intercept at $(0; -11)$).Quadratic Functions Answer:Turning Point: Since $a = 3$ and $q = -12$, the turning point is $(0; -12)$.Axis of Symmetry: $x = 0$.$x$-intercepts: Set $3x^2 - 12 = 0 implies 3(x^2 - 4) = 0 implies 3(x - 2)(x + 2) = 0$. The intercepts are $(-2; 0)$ and $(2; 0)$.Hyperbolic Functions Answer:Domain: $x in mathbb{R}, x neq 0$Range: $y in mathbb{R}, y neq 3$Quadrants: Since $a = -5 0$, the branches occupy the 2nd and 4th quadrants.Exponential Functions Answer:$y$-intercept: Set $x = 0 implies y = 2 cdot 3^0 - 6 = 2(1) - 6 = -4$ (Coordinate $(0; -4)$).Horizontal Asymptote: $y = -6$.Trigonometric Functions Answer:Amplitude: $vert{}-4vert{} = 4$ (Amplitude is always expressed as a positive scalar distance).Range: $y in [-4; 4]$.Graphical Intersections Answer:Equate the two functions: $-x^2 + 9 = 5$Rearrange terms: $-x^2 = 5 - 9 implies -x^2 = -4 implies x^2 = 4$Solve for $x$: $x = 2$ or $x = -2$Substitute back into $g(x) = 5$ to find corresponding $y$-values: $y = 5$ for both.Points of Intersection: $(-2; 5)$ and $(2; 5)$.