Final Review: Graphs
• Graphing the solutions of inequalities in one variable.
x<a
x6a
x>a
x>a
• Rectangular coordinate system
An empty coordinate system:
– the correspondence between points and pairs of real numbers (find the coordinates, plot the
point)
– distance formula: for points (x1 , y1 ) and (x2 , y2 ), the distance d is
d=
p
(x1 − x2 )2 + (y1 − y2 )2
– midpoint formula: for points (x1 , y1 ) and (x2 , y2 ), the midpoint is
¶
µ
x1 + x2 y1 + y2
,
2
2
• Linear equations
– the correspondence between lines and linear equations in two variables
– the slope m of a line passing through two points (x1 , y1 ) and (x2 , y2 ) is given by
m=
y1 − y2
x1 − x2
special case: the slope of a vertical line is undefined
the slopes of parallel lines are the same
the product of slopes of perpendicular lines is −1
– the x-intercept is the point (a, 0) lying on the line or simply a
special case: the x-intercept of a horizontal line is undefined
– the y-intercept is the point (0, b) lying on the line or simply b
special case: the y-intercept of a vertical line is undefined
– the slope-intercept form of a non-vertical line is
y = mx + b
where m is the slope and b is the y-intercept
– the point-slope form of a non-vertical line is
y − y1 = m(x − x1 )
where m is the slope and (x1 , y1 ) is a point on the line
– the equation of a vertical line is x = a, where a is the x-intercept
General steps to find the equation of a line:
1. Find the slope m (maybe it’s given in the problem). If the denominator is 0, the line is a
vertical line.
2. Pick a point given in the problem and use the point-slope form.
3. Simplify to the slope-intercept form if it’s required.
General steps to graph a line from a linear equation:
1. Find two points that satisfy this equation (usually x-intercept and y-intercept).
2. Plot these two points on the coordinate system.
3. Connect them by a line.
• Functions
– the graph of a function satisfies the vertical line test: any vertical line will intersect the graph
at at most one point
– the graph of a one-to-one function satisfies the horizontal line test: any horizontal line will
intersect the graph at at most one point
– transformations of f (x)
shift c
shift c
shift c
shift c
reflect
reflect
reflect
units to the right:
units to the left:
units upwards:
units downwards:
on x-axis:
on y-axis:
on the line y = x:
f (x − c)
f (x + c)
f (x) + c
f (x) − c
−f (x)
f (−x)
f −1 (x)
• Quadratic functions
– the correspondence between parabolas and quadratic functions a(x − h)2 + k
– open upwards if a > 0, downwards if a < 0
– vertex (h, k): lowest point if a > 0, highest point if a < 0
– axis x = h: the parabola is symmetry around this axis
– x-intercepts are the solutions of a(x − h)2 + k = 0
• Exponential functions and logarithmic functions
The asymptote is x-axis.