Topics in Algebra: Graphing

This lecture explains the coordinate plane, including the x and y axes, the origin, and four quadrants, and shows how ordered pairs plot points using left-right and up-down directions.

Determine whether an ordered pair (x, y) solves a linear equation by substituting x first, then y, and applying order of operations to check if both sides match.

Graph linear equations using a modified table of values by substituting x values to find y, plot the coordinates, and draw the line through the points.

Learn how to identify x and y intercepts for linear graphs, with x intercept when y=0 and y intercept when x=0, illustrated by (2,0) and (0,-3) from 3x-2y=6.

Graph linear equations using the intercept method by finding the x-intercept (y=0) and y-intercept (x=0), then plot these two points to draw the line.

Learn slope as rise over run, with m as the slope and change in y over change in x, using positive and negative examples like 2/3, -3/1, and 3/2.

Master slope concepts by using the slope formula m = (y2 - y1)/(x2 - x1) with practical examples, substitutions, and handling negative coordinates to find rise over run.

Learn how slope behaves in special cases: vertical lines give undefined slope, horizontal lines give zero; use two points or the slope formula to compute rise over run.

Graph lines by plotting a point and using rise over run to move, including negative slope, then connect dots to form the line.

Graph linear equations by converting to y = mx + b, where m is the coefficient of x and b is the y-intercept; start at (0, b).

Explore writing lines with slope-intercept and point-slope forms, using parallel and perpendicular relationships and given points to determine slopes and intercepts.

Rewrite each equation into slope intercept form to find the slope of the lines. Compare slopes to determine if the lines are parallel, since parallel lines share the same slope.

Explore perpendicular lines by analyzing slopes, which are inverses and reciprocals with opposite signs, forming right angles. Practice converting equations to slope-intercept form to determine perpendicularity.

Learn to write line equations using point-slope form and convert to slope-intercept form. The lesson covers computing slope from two points, choosing a convenient point, and handling parallel lines.

Graph linear inequalities on the coordinate plane using slope-intercept form. Draw a boundary line (solid for =, dashed for < or >) and shade the appropriate region.

Apply the average rate of change as the slope of a line connecting two points on a function. Use (y2−y1)/(x2−x1) to compute slopes over intervals.

Learn to compute distance between points using the distance formula, substitute coordinates, simplify radicals, and practice with example problems.

Apply the midpoint formula to find the exact midpoint between two coordinates by averaging x and y values, with step-by-step substitution and practice problems.

Identify the domain and range from graphs by tracing x-values left to right. Trace y-values bottom to top and note solid dots for included endpoints; use unions for piecewise graphs.

Identify values that make the denominator zero to determine the domain of rational expressions. Apply factoring and the zero product rule to solve for these domain restrictions and examples.

Explain the domain of complex fractions by solving two denominator constraints, identifying excluded values, and expressing the domain in interval notation with multiple unions.

Apply slope intercept form to word problems by identifying the y intercept and slope. Graph linear relations, such as a $5 start with $1 per ride and taxi costs.

Apply the h s r v h transformation order on the coordinate plane, using horizontal and vertical translations to shift graphs and distinguish shifts affecting x versus the entire function.

Explore transformations on the coordinate plane by applying stretches and shrinks to x or y values and performing reflections over the x- or y-axis, with stepwise examples using f(x)=√x.

Learn to apply the h s r v order for multiple transformations—horizontal translation, stretching, reflection, and vertical translation—on graphs on the coordinate plane, with step-by-step examples.