Graphs and Curves – Comprehensive Study Notes (Straight Lines, Parabolas, Regions, and Curves)
Graphs of Straight Lines
A graph of a linear equation is a straight line.
Form and meaning:
Equation: y = mx + c
m is the gradient (slope) of the line.
c is the y-intercept (the value of y when x = 0).
Key relationships:
Gradient from two points: m = \dfrac{y2 - y1}{x2 - x1}.
x-intercept occurs where y = 0; solve 0 = mx + c to get x = -\dfrac{c}{m} (provided m \neq 0).
y-intercept is simply c (the point where the line crosses the y-axis).
How to read from the equation to sketch:
Identify the gradient m and intercept c from y = mx + c.
Plot the y-intercept at (0, c).
Use the gradient to plot another point: rise by m units for every run of 1 unit in x.
Draw the straight line through these points and extend it across the axes.
Examples (illustrative forms used in the transcript):
Lines of the form y = mx + c with various m and c (e.g., m ∈ {1, 2, -1, -2, 3} and c as an integer) appear throughout the exercises.
Specific example forms you might encounter include:
y = x + 3\,;\, y = x + 5\,;\, y = x - 3
y = 2x + 1\,;\, y = -x + 3\,;\, y = -2x + 1
other lines with coefficients like 3x, -2x, etc.
Inequalities and shading (intro):
When solving problems that involve shading a region, you typically have inequalities of the form:
y > mx + c (region above the line) or y < mx + c (region below the line).
If the inequality is non-strict ((\ge) or (\le)), the boundary line is included (solid line). If it is strict ((>) or (<)), the boundary line is dashed.
Three-line region (region R):
Often you are asked to shade the region bounded by three lines.
The region is the intersection of three half-planes, each defined by an inequality of the form ax + by \le c or similar.
You are asked to write down the three inequalities that define R and identify the region labeled R on the diagram.
Graphs of Straight Lines: Practice-style questions from the transcript
Sketch and interpret lines given by equations such as: y = mx + c with several values of m and c.
Shade regions for inequalities: examples include
y > x + 2
y > 2x + 1
y < 3x + 2
y < -2x + 1
Solve by graphing: find the region(s) that satisfy multiple inequalities simultaneously.
Reading off intercepts from a graph and converting between intercept form and gradient-intercept form.
Graphs of Curves (Parabolas)
A quadratic graph has the form y = ax^2 + bx + c.
Shape:
If a > 0, the parabola opens upwards and has a minimum at the vertex.
If a < 0, the parabola opens downwards and has a maximum at the vertex.
Vertex and axis of symmetry:
Vertex x-coordinate: x_v = -\dfrac{b}{2a}
Vertex y-coordinate: yv = f(xv) = a xv^2 + b xv + c
Axis of symmetry is the vertical line x = x_v.
Intercepts and roots:
x-intercepts are solutions to ax^2 + bx + c = 0.
Discriminant: \Delta = b^2 - 4ac
If \Delta > 0 there are two real roots.
If \Delta = 0 there is one real root (the parabola touches the x-axis).
If \Delta < 0 there are no real x-intercepts.
Vertex value and minimum/maximum:
For a > 0, the minimum value occurs at the vertex.
For a < 0, the maximum value occurs at the vertex.
Example family of tasks from the transcript (parabola-themed):
Determine the vertex and sketch the parabola for given coefficients.
Find and plot the x-intercepts and y-intercept.
Use completing the square or discriminant to discuss the number of real roots.
Solve equations graphically by finding intersection points with the x-axis or other curves.
Graphs of Curves: Specific function forms (beyond quadratics)
Trigonometric forms:
General sine/cosine forms used in the transcript include:
y = a\sin(x) + b
y = a\cos(x + b) (degrees were used in some problems; you may see x in degrees in the exam material)
Characteristics:
Amplitude: |a| (peak-to-trough height is 2|a| if unshifted)
Vertical shift: b shifts the graph up or down
Phase shift: inside the argument (e.g., \cos(x + b)) shifts left/right
Period depends on the coefficient of the angle (for \sin(kx) or \cos(kx), period = \dfrac{2\pi}{k} radians or \dfrac{360^{\circ}}{k} degrees)
Graphs of higher-order and composed functions appear in the transcript’s practice (e.g., cubic, quartic forms, and combinations such as y = ax^3 - 3x + 1 or y = x^2 - 4x + 3).
How to approach curve graphs:
Sketch key features: intercepts, turning points, symmetry, and end behavior if applicable.
For trig graphs, identify amplitude, period, phase shift, and vertical shift.
For polynomial graphs, locate critical points and approximate zeros via the discriminant or root-finding methods when solving graphically.
Simultaneous Equations and Regions (Graphical methods)
Graphical solution of simultaneous equations:
Plot each equation as a line (or curve for non-linear equations) on the same axes.
The solution is the intersection point(s) of the graphs.
Example: Solve systems of the form y = mx + c and another line; the intersection provides the solution (x, y).
Regions defined by multiple inequalities:
The region bounded by multiple lines is the intersection of the corresponding half-planes.
To define the region with inequalities, you write one inequality per boundary line and specify the side that is shaded.
Typical exam task (as in the transcript):
Given a region R bounded by three straight lines with equations (or inequalities), write down the three inequalities that define R and shade accordingly.
Example shading tasks include ensuring the region satisfies all three inequalities simultaneously.
Sketched Practice Problems (conceptual overview drawn from the transcript)
Graphs to sketch include:
Graphs of the form y = mx + c for various m and c to reinforce understanding of slope and intercepts.
Parabolas y = ax^2 + bx + c with varying a, b, c to identify vertex, axis of symmetry, and intercepts.
Linear inequalities and the shaded regions they define, including cases with three lines bounding a polygonal region.
Curves such as y = a\sin x + b and y = a\cos(x + b) to reinforce amplitude, period, and phase shift concepts.
Graphs of higher-order polynomials and functions such as y = x^3 - 3x + 1, plotted for integer x values to build intuition for turning points and intercepts.
World-building and cross-connections:
Intercept-intercept consistency helps check sketch accuracy by confirming intercepts align with the equation.
Understanding how changing a, b, or c alters the graph (steepness, shift, and vertical placement) aids rapid problem solving under exam conditions.
Graphical solutions provide visual confirmation for algebraic methods (substitution, elimination) in simultaneous equations.
Quick-reference formulas (for review)
Linear line:
y = mx + c
m = \dfrac{y2 - y1}{x2 - x1}
Intercepts: y-intercept = c; x-intercept = -\dfrac{c}{m} (if m \neq 0)
Parabola (quadratic):
y = ax^2 + bx + c
Vertex: xv = -\dfrac{b}{2a},\quad yv = a xv^2 + b xv + c
Axis of symmetry: x = x_v
Opening: a > 0 \Rightarrow \text{min}; a < 0 \Rightarrow \text{max}
Discriminant: \Delta = b^2 - 4ac
\Delta > 0: two real roots
\Delta = 0: one real root (tangent)
\Delta < 0: no real roots
Linear inequalities for shading:
Boundary: either dashed (strict) or solid (non-strict) depending on the symbol used.
Regions above/below lines determined by the sign in y > mx + c or y < mx + c.
Trigonometric prototypes (idea):
y = a\sin(x) + b; amplitude |a|, vertical shift b, period affected by any coefficient inside the sine.
y = a\cos(x + b); amplitude |a|, phase shift -b, period as above.
Intersections and regions:
Graphical intersection gives solution to simultaneous equations.
Region defined by multiple lines is the intersection of the corresponding half-planes.
Tips for exam preparation
Practice sketching quickly: identify gradient and intercept from the equation, then plot two points and draw the line.
For parabolas, always determine the vertex first; it helps locate the turning point and the axis of symmetry.
When shading inequalities, decide boundary style (solid/dashed) based on whether inequalities are non-strict or strict.
In region problems, label the region and ensure the inequalities collectively describe the shaded area.
For trigonometric graphs, note the amplitude, period, and shifts to quickly sketch and compare with standard sine/cosine waves.
Use graphing as a cross-check for algebraic methods when solving systems or inequalities.
Quick reference: notational reminders from the transcript
y denotes the vertical axis value, x denotes the horizontal axis value.
The transcript uses the standard Cartesian coordinate setup; many problems are presented as sketches on grids with marked axes and intercepts.
It includes a large set of practice problems labeled by pages and question numbers, illustrating a broad range of graphing and shading tasks, from simple straight lines to complex regions bounded by multiple lines and curves.
Many questions require you to read off from graphs, interpolate/extrapolate values, or determine approximate solutions by drawing additional lines or estimating intersections.