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Intermediate Algebra

Key Concepts

Intermediate AlgebraKey Concepts

Key Concepts

3.1 Graph Linear Equations in Two Variables

  • Points on the Axes
    • Points with a y-coordinate equal to 0 are on the x-axis, and have coordinates (a,0).(a,0).
    • Points with an x-coordinate equal to 00 are on the y-axis, and have coordinates (0,b).(0,b).
  • Quadrant
    Quadrant IQuadrant IIQuadrant IIIQuadrant IV (x,y)(x,y)(x,y)(x,y) (+,+)(−,+)(−,−)(+,−) Quadrant IQuadrant IIQuadrant IIIQuadrant IV (x,y)(x,y)(x,y)(x,y) (+,+)(−,+)(−,−)(+,−)
    This figure shows the x y-coordinate plane with the four quadrants labeled. In the top right of the plane is quadrant I labeled (plus, plus). In the top left of the plane is quadrant II labeled (minus, plus). In the bottom left of the plane is quadrant III labeled (minus, minus). In the bottom right of the plane is quadrant IV labeled (plus, minus).
  • Graph of a Linear Equation: The graph of a linear equation Ax+By=CAx+By=C is a straight line.
     Every point on the line is a solution of the equation.
     Every solution of this equation is a point on this line.
  • How to graph a linear equation by plotting points.
    1. Step 1. Find three points whose coordinates are solutions to the equation. Organize them in a table.
    2. Step 2. Plot the points in a rectangular coordinate system. Check that the points line up. If they do not, carefully check your work.
    3. Step 3. Draw the line through the three points. Extend the line to fill the grid and put arrows on both ends of the line.
  • x-intercept and y-intercept of a Line
    • The x-intercept is the point (a,0)(a,0) where the line crosses the x-axis.
    • The y-intercept is the point (0,b)(0,b) where the line crosses the y-axis. The table has 3 rows and 2 columns. The first row is a header row with the headers x and y. The second row contains a and 0. The x-intercept occurs when y is zero. The third row contains 0 and b. The y-intercept occurs when x is zero.
  • Find the x- and y-intercepts from the Equation of a Line
    • Use the equation of the line. To find:
       the x-intercept of the line, let y=0y=0 and solve for x.
       the y-intercept of the line, let x=0x=0 and solve for y.
  • How to graph a linear equation using the intercepts.
    1. Step 1. Find the x- and y-intercepts of the line.
       Let y=0y=0 and solve for x.
       Let x=0x=0 and solve for y.
    2. Step 2. Find a third solution to the equation.
    3. Step 3. Plot the three points and check that they line up.
    4. Step 4. Draw the line

3.2 Slope of a Line

  • Slope of a Line
    • The slope of a line is m=riserun.m=riserun.
    • The rise measures the vertical change and the run measures the horizontal change.




  • How to find the slope of a line from its graph using m=riserun.m=riserun.
    1. Step 1. Locate two points on the line whose coordinates are integers.
    2. Step 2. Starting with one point, sketch a right triangle, going from the first point to the second point.
    3. Step 3. Count the rise and the run on the legs of the triangle.
    4. Step 4. Take the ratio of rise to run to find the slope: m=riserun.m=riserun.
  • Slope of a line between two points.
    • The slope of the line between two points (x1,y1)(x1,y1) and (x2,y2)(x2,y2) is:
      m=y2−y1x2−x1.m=y2−y1x2−x1.
  • How to graph a line given a point and the slope.
    1. Step 1. Plot the given point.
    2. Step 2. Use the slope formula m=riserunm=riserun to identify the rise and the run.
    3. Step 3. Starting at the given point, count out the rise and run to mark the second point.
    4. Step 4. Connect the points with a line.
  • Slope Intercept Form of an Equation of a Line
    • The slope–intercept form of an equation of a line with slope m and y-intercept, (0,b)(0,b) is y=mx+by=mx+b The table has a title row that reads “Methods to Graph Lines”. Below this are four columns. The first column contains the following: Point Plotting. A blank table with two columns and four rows. The first row is a header row with the headers “x” and “y”. Find three points. Plot the points, make sure they line up, them draw the line. The second column contains: Slope-Intercept. Y equals m x plus b. Find the slope and y-intercept, then count the slope to get a second point. The third column: Intercepts. A table with two columns and four rows. The first row is a header row with the headers “x” and “y”. In the first row there is a 0 in the x column. In the second row there is a 0 in the y column. The remaining spaces are blank. Fourth column. Recognize vertical and horizontal lines. The equation has only one variable. X equals a vertical. Y equals b horizontal.
  • Parallel Lines
    • Parallel lines are lines in the same plane that do not intersect.
       Parallel lines have the same slope and different y-intercepts.
       If m1m1 and m2m2 are the slopes of two parallel lines then m1=m2.m1=m2.
       Parallel vertical lines have different x-intercepts.
  • Perpendicular Lines
    • Perpendicular lines are lines in the same plane that form a right angle.
    • If m1m1 and m2m2 are the slopes of two perpendicular lines, then:
       their slopes are negative reciprocals of each other, m1=−1m2.m1=−1m2.
       the product of their slopes is −1,−1,m1·m2=−1.m1·m2=−1.
    • A vertical line and a horizontal line are always perpendicular to each other.

3.3 Find the Equation of a Line

  • How to find an equation of a line given the slope and a point.
    1. Step 1. Identify the slope.
    2. Step 2. Identify the point.
    3. Step 3. Substitute the values into the point-slope form, y−y1=m(x−x1).y−y1=m(x−x1).
    4. Step 4. Write the equation in slope-intercept form.



  • How to find an equation of a line given two points.
    1. Step 1. Find the slope using the given points. m=y2−y1x2−x1m=y2−y1x2−x1
    2. Step 2. Choose one point.
    3. Step 3. Substitute the values into the point-slope form: y−y1=m(x−x1).y−y1=m(x−x1).
    4. Step 4.
      Write the equation in slope-intercept form.
      To Write an Equation of a Line
      If given: Use: Form:
      Slope and y-intercept slope-intercept y=mx+by=mx+b
      Slope and a point point-slope y−y1=m(x−x1)y−y1=m(x−x1)
      Two points point-slope y−y1=m(x−x1)y−y1=m(x−x1)
  • How to find an equation of a line parallel to a given line.
    1. Step 1. Find the slope of the given line.
    2. Step 2. Find the slope of the parallel line.
    3. Step 3. Identify the point.
    4. Step 4. Substitute the values into the point-slope form: y−y1=m(x−x1).y−y1=m(x−x1).
    5. Step 5. Write the equation in slope-intercept form
  • How to find an equation of a line perpendicular to a given line.
    1. Step 1. Find the slope of the given line.
    2. Step 2. Find the slope of the perpendicular line.
    3. Step 3. Identify the point.
    4. Step 4. Substitute the values into the point-slope form, y−y1=m(x−x1)y−y1=m(x−x1)
    5. Step 5. Write the equation in slope-intercept form.

3.4 Graph Linear Inequalities in Two Variables

  • How to graph a linear inequality in two variables.
    1. Step 1. Identify and graph the boundary line.
       If the inequality is ≤or≥,≤or≥, the boundary line is solid.
       If the inequality is <or>,<or>, the boundary line is dashed.
    2. Step 2. Test a point that is not on the boundary line. Is it a solution of the inequality?
    3. Step 3. Shade in one side of the boundary line.
       If the test point is a solution, shade in the side that includes the point.
       If the test point is not a solution, shade in the opposite side.

3.5 Relations and Functions

  • Function Notation: For the function y=f(x)y=f(x)
    • f is the name of the function
    • x is the domain value
    • f(x)f(x) is the range value y corresponding to the value x
       We read f(x)f(x) as f of x or the value of f at x.
  • Independent and Dependent Variables: For the function y=f(x),y=f(x),
    • x is the independent variable as it can be any value in the domain
    • y is the dependent variable as its value depends on x

3.6 Graphs of Functions

  • Vertical Line Test
    • A set of points in a rectangular coordinate system is the graph of a function if every vertical line intersects the graph in at most one point.
    • If any vertical line intersects the graph in more than one point, the graph does not represent a function.
  • Graph of a Function
    • The graph of a function is the graph of all its ordered pairs, (x,y)(x,y) or using function notation, (x,f(x))(x,f(x)) where y=f(x).y=f(x).
      fname of functionxx-coordinate of the ordered pairf(x)y-coordinate of the ordered pairfname of functionxx-coordinate of the ordered pairf(x)y-coordinate of the ordered pair
  • Linear Function
     This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsm x plus b”, “m, b: all real numbers”, “m: slope of the line”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.
  • Constant Function
     This figure has a graph of a straight horizontal line on the x y-coordinate plane. The line goes through the point (0, b). Next to the graph are the following: “f of x equalsb”, “b: any real number”, “b: y-intercept”, “Domain: (negative infinity, infinity)”, and “Range: b”.
  • Identity Function
     This figure has a graph of a straight line on the x y-coordinate plane. The line goes through the points (0, 0), (1, 1), and (2, 2). Next to the graph are the following: “f of x equalsx”, “m: 1”, “b: 0”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.
  • Square Function
     This figure has a graph of a parabola opening up graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 2 to 6. The parabola goes through the points (negative 2, 4), (negative 1, 1), (0, 0), (1, 1), and (2, 4). Next to the graph are the following: “f of x equalsx squared”, “Domain: (negative infinity, infinity)”, and “Range: [0, infinity)”.
  • Cube Function
     This figure has a curved line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 4 to 4. The curved line goes through the points (negative 2, negative 8), (negative 1, negative 1), (0, 0), (1, 1), and (2, 8).). Next to the graph are the following: “f of x equalsx cubed”, “Domain: (negative infinity, infinity)”, and “Range: (negative infinity, infinity)”.
  • Square Root Function
     This figure has a curved half-line graphed on the x y-coordinate plane. The x-axis runs from 0 to 8. The y-axis runs from 0 to 8. The curved half-line starts at the point (0, 0) and then goes up and to the right. The curved half line goes through the points (1, 1) and (4, 2). Next to the graph are the following: “f of x equalssquare root of x”, “Domain: [0, infinity)”, and “Range: [0, infinity)”.
  • Absolute Value Function
     This figure has a v-shaped line graphed on the x y-coordinate plane. The x-axis runs from negative 4 to 4. The y-axis runs from negative 1 to 6. The v-shaped line goes through the points (negative 3, 3), (negative 2, 2), (negative 1, 1), (0, 0), (1, 1), (2, 2), and (3, 3). The point (0, 0) where the line changes slope is called the vertex. Next to the graph are the following: “f of x equalsabsolute value of x”, “Domain: (negative infinity, infinity)”, and “Range: [0, infinity)”.
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