{"id":9886,"date":"2026-07-03T17:39:42","date_gmt":"2026-07-03T17:39:42","guid":{"rendered":"https:\/\/kapdec.com\/help\/?p=9886"},"modified":"2026-07-03T17:39:42","modified_gmt":"2026-07-03T17:39:42","slug":"linear-functions-in-a-coordinate-planes","status":"publish","type":"post","link":"https:\/\/kapdec.com\/help\/linear-functions-in-a-coordinate-planes\/","title":{"rendered":"Linear Functions In A Coordinate Planes"},"content":{"rendered":"
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KAPDEC® | Elite STEM Learning Platform | https:\/\/kapdec.com<\/a><\/p>\n

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Unit: <\/strong>Functions<\/strong><\/h2>\n

Chapter: <\/strong>Linear Functions in a Coordinated Plane<\/strong><\/h3>\n

Reference: – What is a Linear Function, Slope (Rate of Change), y-intercept, Slope-Intercept Form (y = mx + b), Graphing Linear Functions, Horizontal and Vertical Lines, Standard Form (Ax + By = C), Point-Slope Form, Finding Slope from Two Points, Interpreting Slope and y-intercept in Real Life, Comparing Linear Functions, Solved Examples, Odd-One-Out Problems, Common Mistakes<\/em><\/p>\n

After studying this chapter, you should be able to understand:<\/strong><\/p>\n

    \n
  • What is a Linear Function and How to Identify It<\/em><\/li>\n
  • Slope as Rate of Change<\/em><\/li>\n
  • Graphing Linear Functions Using Slope and y-intercept<\/em><\/li>\n
  • Writing Equations in Slope-Intercept Form<\/em><\/li>\n
  • Understanding Standard Form and Point-Slope Form<\/em><\/li>\n
  • Interpreting Linear Functions in Real-World Contexts<\/em><\/li>\n<\/ul>\n

    Introduction to Linear Functions in a Coordinate Plane<\/strong><\/p>\n

    Definition<\/u><\/strong><\/p>\n

    A linear function is a function whose graph is a straight line. It can be written in the form f(x) = mx + b, where m is the slope (rate of change) and b is the y-intercept (where the line crosses the y-axis).<\/p>\n

    In a coordinate plane, every linear function represents a straight line. The line shows all the ordered pairs (x, y) that satisfy the equation.<\/p>\n

    When we study linear functions in a coordinate plane, we essentially ask:<\/p>\n

    “How does the line look on the graph? How steep is it? Where does it cross the axes?”<\/p>\n

    Once we understand these features, we can graph any linear function quickly and interpret what it means in real life.<\/p>\n

    Importance of Linear Functions<\/u><\/strong><\/p>\n

      \n
    • Models constant rate situations (speed, cost per item, temperature change)<\/li>\n
    • Foundation for understanding more complex functions<\/li>\n
    • Used in economics (supply and demand), physics (motion), and business (profit)<\/li>\n
    • Helps make predictions based on patterns<\/li>\n
    • Essential for understanding data trends and scatter plots<\/li>\n<\/ul>\n

      Example<\/strong><\/p>\n

      The function f(x) = 2x + 1 is linear.
      \nIts graph is a straight line with slope 2 (goes up 2 units for every 1 unit right) and crosses the y-axis at (0, 1).
      \nSo, if we have f(x) = x\u00b2, that is NOT linear (the graph is a curve).<\/p>\n

      \u00a0<\/p>\n

      Subtopics<\/u><\/strong><\/p>\n

      1. Slope (Rate of Change)<\/strong><\/p>\n

      The slope of a line measures how steep the line is. It tells us how much y changes when x increases by 1. Slope is often called the “rate of change.”<\/p>\n

      Formula for Slope:<\/strong>
      \nGiven two points (x\u2081, y\u2081) and (x\u2082, y\u2082) on a line, the slope m is:<\/p>\n

      m = (y\u2082 – y\u2081) \/ (x\u2082 – x\u2081)<\/p>\n

      Ways to Think About Slope:<\/strong><\/p>\n

      Slope = rise \/ run
      \nRise is the vertical change (how many units up or down)
      \nRun is the horizontal change (how many units right)<\/p>\n

      Examples of Slope:<\/strong><\/p>\n

      If m = 2, the line goes up 2 units for every 1 unit right (steep upward)<\/p>\n

      If m = 1\/2, the line goes up 1 unit for every 2 units right (gentle upward)<\/p>\n

      If m = 0, the line is horizontal (flat, no rise)<\/p>\n

      If m = -3, the line goes down 3 units for every 1 unit right (steep downward)<\/p>\n

      If m = -1\/4, the line goes down 1 unit for every 4 units right (gentle downward)<\/p>\n

      Special Slopes:<\/strong><\/p>\n

      Positive slope (m > 0) \u2013 line rises from left to right<\/p>\n

      Negative slope (m < 0) \u2013 line falls from left to right<\/p>\n

      Zero slope (m = 0) \u2013 horizontal line<\/p>\n

      Undefined slope \u2013 vertical line (the run is zero, so we cannot divide by zero)<\/p>\n

      2. y-intercept<\/strong><\/p>\n

      The y-intercept is the point where the line crosses the y-axis. At this point, x = 0. The y-intercept is written as (0, b), where b is the y-coordinate.<\/p>\n

      Example:<\/strong>\u00a0In f(x) = 3x + 5, the y-intercept is 5, so the line crosses the y-axis at (0, 5).<\/p>\n

      Why the y-intercept matters:<\/strong>\u00a0It tells us the starting value or initial amount in real-world problems. For example, if a taxi charge\u00a02 flat fee, the y-intercept (2) is the flat fee.<\/p>\n

      3. Slope-Intercept Form (y = mx + b)<\/strong><\/p>\n

      This is the most useful form for graphing linear functions quickly.<\/p>\n

      General Form:<\/strong>\u00a0y = mx + b
      \nm = slope
      \nb = y-intercept<\/p>\n

      Example 1:<\/strong>\u00a0y = 2x – 3
      \nSlope m = 2, y-intercept b = -3 (crosses y-axis at (0, -3))<\/p>\n

      Example 2:<\/strong>\u00a0y = -4x + 7
      \nSlope m = -4, y-intercept b = 7<\/p>\n

      Example 3:<\/strong>\u00a0y = x + 2 (here m = 1, b = 2)<\/p>\n

      Example 4:<\/strong>\u00a0y = 5 (here m = 0, b = 5 \u2013 horizontal line)<\/p>\n

      How to Graph a Line Using Slope-Intercept Form:<\/strong><\/p>\n

      Step 1: Plot the y-intercept (0, b) on the y-axis.<\/p>\n

      Step 2: Use the slope m = rise\/run to find another point. From the y-intercept, go up (if positive rise) or down (if negative rise) and then right.<\/p>\n

      Step 3: Draw a straight line through the two points.<\/p>\n

      Example \u2013 Graph y = 2x + 1<\/strong><\/p>\n

      Step 1: y-intercept is 1 \u2192 plot (0, 1)<\/p>\n

      Step 2: Slope is 2 = 2\/1, so rise = 2, run = 1. From (0, 1), go up 2 and right 1 \u2192 reach (1, 3)<\/p>\n

      Step 3: Draw the line through (0, 1) and (1, 3)<\/p>\n

      Example \u2013 Graph y = -1\/2 x + 3<\/strong><\/p>\n

      Step 1: y-intercept is 3 \u2192 plot (0, 3)<\/p>\n

      Step 2: Slope is -1\/2 = -1\/2, so rise = -1, run = 2. From (0, 3), go down 1 and right 2 \u2192 reach (2, 2)<\/p>\n

      Step 3: Draw the line through (0, 3) and (2, 2)<\/p>\n

      4. Horizontal and Vertical Lines<\/strong><\/p>\n

      Horizontal Lines:<\/strong>
      \nEquation: y = c (where c is a constant)
      \nSlope m = 0
      \ny-intercept = c
      \nGraph is a flat line crossing the y-axis at (0, c)<\/p>\n

      Example:<\/strong>\u00a0y = 4 is a horizontal line through (0, 4). Every point on this line has y = 4.<\/p>\n

      Vertical Lines:<\/strong>
      \nEquation: x = c (where c is a constant)
      \nSlope is undefined
      \nNo y-intercept (unless c = 0, then it’s the y-axis itself)
      \nGraph is a straight up-and-down line crossing the x-axis at (c, 0)<\/p>\n

      Example:<\/strong>\u00a0x = -2 is a vertical line through (-2, 0). Every point on this line has x = -2.<\/p>\n

      Important:<\/strong>\u00a0A vertical line is NOT a function because one input (x = c) gives infinitely many outputs. But we still study it in the coordinate plane.<\/p>\n

      5. Standard Form (Ax + By = C)<\/strong><\/p>\n

      Another way to write linear equations is standard form: Ax + By = C, where A, B, and C are integers, and A is usually positive.<\/p>\n

      Example:<\/strong>\u00a03x + 2y = 6 is in standard form.<\/p>\n

      To graph from standard form:<\/strong>
      \nMethod 1 \u2013 Convert to slope-intercept form by solving for y.
      \nMethod 2 \u2013 Find the x-intercept (set y = 0) and y-intercept (set x = 0), then plot.<\/p>\n

      Example \u2013 Graph 2x + y = 4<\/strong><\/p>\n

      Method 1: Solve for y: y = -2x + 4 \u2192 slope = -2, y-intercept = 4<\/p>\n

      Method 2: Find intercepts \u2013
      \nx-intercept: set y=0 \u2192 2x = 4 \u2192 x = 2 \u2192 (2, 0)
      \ny-intercept: set x=0 \u2192 y = 4 \u2192 (0, 4)
      \nPlot (2,0) and (0,4) and draw the line.<\/p>\n

      6. Point-Slope Form<\/strong><\/p>\n

      When you know the slope and one point on the line, use point-slope form: y – y\u2081 = m(x – x\u2081)<\/p>\n

      Example:<\/strong>\u00a0A line has slope 3 and passes through (2, 5).
      \nEquation: y – 5 = 3(x – 2)
      \nSimplify: y – 5 = 3x – 6 \u2192 y = 3x – 1<\/p>\n

      When to use point-slope form:<\/strong>\u00a0When you are given a slope and a point, or two points (find slope first, then use one point).<\/p>\n

      7. Finding Slope from Two Points<\/strong><\/p>\n

      Given two points (x\u2081, y\u2081) and (x\u2082, y\u2082), use the slope formula: m = (y\u2082 – y\u2081) \/ (x\u2082 – x\u2081)<\/p>\n

      Example 1:<\/strong>\u00a0Find slope between (2, 3) and (5, 9)
      \nm = (9 – 3) \/ (5 – 2) = 6 \/ 3 = 2<\/p>\n

      Example 2:<\/strong>\u00a0Find slope between (4, 7) and (4, 10)
      \nm = (10 – 7) \/ (4 – 4) = 3 \/ 0 = undefined (vertical line)<\/p>\n

      Example 3:<\/strong>\u00a0Find slope between (2, 5) and (6, 5)
      \nm = (5 – 5) \/ (6 – 2) = 0 \/ 4 = 0 (horizontal line)<\/p>\n

      8. Interpreting Slope and y-intercept in Real Life<\/strong><\/p>\n

      In real-world situations, the slope represents the rate of change, and the y-intercept represents the starting value.<\/p>\n

      Example 1 \u2013 Car Rental:<\/strong>
      \nA car rental company charges\u00a050 one-time fee.
      \nEquation: C = 30d + 50, where C is total cost and d is number of days.
      \nSlope = 30 (cost increases by $30 per day)
      \ny-intercept = 50 (the initial fee before any days)<\/p>\n

      Example 2 \u2013 Water Tank:<\/strong>
      \nA water tank has 200 gallons and is draining at 5 gallons per minute.
      \nEquation: W = -5t + 200, where W is water in gallons and t is time in minutes.
      \nSlope = -5 (water decreases by 5 gallons each minute)
      \ny-intercept = 200 (starting amount of water)<\/p>\n

      Example 3 \u2013 Earnings:<\/strong>
      \nA worker earns\u00a020 bonus.
      \nEquation: E = 15h + 20, where E is earnings and h is hours worked.
      \nSlope = 15 (earns $15 per hour)
      \ny-intercept = 20 (bonus earned even with zero hours)<\/p>\n

      \u00a0<\/p>\n

      Solved Examples<\/strong><\/p>\n

      Example 1:<\/strong>\u00a0Find the slope and y-intercept of y = -3x + 7.<\/p>\n

      Solution:<\/strong>\u00a0The equation is in slope-intercept form y = mx + b. Here m = -3 and b = 7.<\/p>\n

      Answer:<\/strong>\u00a0Slope = -3, y-intercept = 7<\/p>\n

      \u00a0<\/p>\n

      Example 2:<\/strong>\u00a0Graph the function f(x) = (1\/2) – 2.<\/p>\n

      Solution:<\/strong>
      \ny-intercept is -2 \u2192 plot (0, -2)
      \nSlope is 1\/2, so rise = 1, run = 2. From (0, -2), go up 1 and right 2 \u2192 reach (2, -1)
      \nDraw the line through (0, -2) and (2, -1)<\/p>\n

      Answer:<\/strong>\u00a0[Graph description \u2013 line rising gently from left to right crossing y-axis at -2]<\/p>\n

      \u00a0<\/p>\n

      Example 3:<\/strong>\u00a0Find the slope between the points (-3, 2) and (4, -5).<\/p>\n

      Solution:<\/strong>
      \nm = (y\u2082 – y\u2081) \/ (x\u2082 – x\u2081) = (-5 – 2) \/ (4 – (-3)) = (-7) \/ (7) = -1<\/p>\n

      Answer:<\/strong>\u00a0Slope = -1<\/p>\n

      \u00a0<\/p>\n

      Example 4:<\/strong>\u00a0Write the equation of a line with slope 4 that passes through (1, 3).<\/p>\n

      Solution:<\/strong>\u00a0Use point-slope form: y – y\u2081 = m(x – x\u2081)
      \ny – 3 = 4(x – 1)
      \ny – 3 = 4x – 4
      \ny = 4x – 1<\/p>\n

      Answer:<\/strong>\u00a0y = 4x – 1<\/p>\n

      \u00a0<\/p>\n

      Example 5:<\/strong>\u00a0Graph the equation 2x – y = 3.<\/p>\n

      Solution:<\/strong>\u00a0Convert to slope-intercept form:
      \ny = 2x – 3
      \ny-intercept = -3 \u2192 plot (0, -3)
      \nSlope = 2 = 2\/1 \u2192 from (0, -3), go up 2 and right 1 \u2192 reach (1, -1)
      \nDraw the line through (0, -3) and (1, -1)<\/p>\n

      Answer:<\/strong>\u00a0[Graph description \u2013 line rising steeply crossing y-axis at -3]<\/p>\n

      \u00a0<\/p>\n

      Example 6 \u2013 Odd One Out (Slope Values):<\/strong><\/p>\n

      Examine the five slopes described below. Exactly one is different from the others in a significant way. Identify it.<\/strong><\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Item<\/p>\n<\/td>\n

      		\n

      Slope Description<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      A<\/p>\n<\/td>\n

      		\n

      Slope = 0<\/p>\n<\/td>\n<\/tr>\n

      \n

      B<\/p>\n<\/td>\n

      		\n

      Slope = -2<\/p>\n<\/td>\n<\/tr>\n

      \n

      C<\/p>\n<\/td>\n

      		\n

      Slope = 5<\/p>\n<\/td>\n<\/tr>\n

      \n

      D<\/p>\n<\/td>\n

      		\n

      Slope = 1\/3<\/p>\n<\/td>\n<\/tr>\n

      \n

      E<\/p>\n<\/td>\n

      		\n

      Slope = undefined<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Solution:<\/strong><\/p>\n

      Slope = 0 \u2192 horizontal line (A)
      \nSlope = -2 \u2192 negative, steep downward (B)
      \nSlope = 5 \u2192 positive, steep upward (C)
      \nSlope = 1\/3 \u2192 positive, gentle upward (D)
      \nSlope = undefined \u2192 vertical line (E)<\/p>\n

      Three reasons why E is the odd one out:<\/strong><\/p>\n

      (A)<\/strong>\u00a0Slope undefined means the line is vertical, which is NOT a function. All other slopes (0, -2, 5, 1\/3) represent functions.
      \n(B)<\/strong>\u00a0A vertical line cannot be written in slope-intercept form y = mx + b. All others can.
      \n(C)<\/strong>\u00a0The run for a vertical line is zero, making the slope formula undefined due to division by zero. All other slopes have non-zero run.<\/p>\n

      Conclusion:<\/strong>\u00a0E (slope undefined) is the odd one out.<\/p>\n

      \u00a0<\/p>\n

      Example 7 \u2013 Odd One Out (Linear vs non-linear):<\/strong><\/p>\n

      Examine the five equations below. Exactly one is NOT a linear function. Identify it.<\/strong><\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Item<\/p>\n<\/td>\n

      		\n

      Equation<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      P<\/p>\n<\/td>\n

      		\n

      y = 2x – 5<\/p>\n<\/td>\n<\/tr>\n

      \n

      Q<\/p>\n<\/td>\n

      		\n

      y = -x + 3<\/p>\n<\/td>\n<\/tr>\n

      \n

      R<\/p>\n<\/td>\n

      		\n

      y = 4<\/p>\n<\/td>\n<\/tr>\n

      \n

      S<\/p>\n<\/td>\n

      		\n

      y = x\u00b2 + 1<\/p>\n<\/td>\n<\/tr>\n

      \n

      T<\/p>\n<\/td>\n

      		\n

      y = (1\/2)x<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Solution:<\/strong><\/p>\n

      P: y = 2x – 5 \u2192 linear (form y = mx + b)
      \nQ: y = -x + 3 \u2192 linear (m = -1, b = 3)
      \nR: y = 4 \u2192 linear (horizontal line, m = 0, b = 4)
      \nS: y = x\u00b2 + 1 \u2192 NOT linear (x is squared, graph is a parabola)
      \nT: y = (1\/2)x \u2192 linear (m = 1\/2, b = 0)<\/p>\n

      Three reasons why S is the odd one out:<\/strong><\/p>\n

      (A)<\/strong>\u00a0S has x raised to the power 2, making it a quadratic function, not linear. All others have x to the first power only.
      \n(B)<\/strong>\u00a0The graph of S is a parabola (U-shaped curve), while the graphs of P, Q, R, T are straight lines.
      \n(C)<\/strong>\u00a0Linear functions have a constant rate of change (slope). S has a changing rate of change.<\/p>\n

      Conclusion:<\/strong>\u00a0S (y = x\u00b2 + 1) is the odd one out.<\/p>\n

      \u00a0<\/p>\n

      Example 8 \u2013 Odd One Out (Graph Interpretation):<\/strong><\/p>\n

      Examine the five-line descriptions below. Exactly one represents a line that is NOT a function. Identify it.<\/strong><\/p>\n

      \n\n\n\n\n\n\n\n\n\n
      \n

      Item<\/p>\n<\/td>\n

      		\n

      Line Description<\/p>\n<\/td>\n<\/tr>\n<\/thead>\n

      \n

      A<\/p>\n<\/td>\n

      		\n

      Horizontal line through (0, 3)<\/p>\n<\/td>\n<\/tr>\n

      \n

      B<\/p>\n<\/td>\n

      		\n

      Line with slope 0<\/p>\n<\/td>\n<\/tr>\n

      \n

      C<\/p>\n<\/td>\n

      		\n

      Vertical line through (-2, 0)<\/p>\n<\/td>\n<\/tr>\n

      \n

      D<\/p>\n<\/td>\n

      		\n

      Line with equation y = 4x<\/p>\n<\/td>\n<\/tr>\n

      \n

      E<\/p>\n<\/td>\n

      		\n

      Line passing through (1,2) and (3,2)<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

      Source: Kapdec.com<\/p>\n<\/div>\n

      Solution:<\/strong><\/p>\n

      A: Horizontal line through (0,3) \u2192 equation y = 3 \u2192 function \u2713<\/p>\n

      B: Slope 0 \u2192 horizontal line \u2192 function \u2713<\/p>\n

      C: Vertical line through (-2,0) \u2192 equation x = -2 \u2192 NOT a function (fails vertical line test) \u2717<\/p>\n

      D: y = 4x \u2192 linear function \u2713<\/p>\n

      E: Points (1,2) and (3,2) \u2192 horizontal line y = 2 \u2192 function \u2713<\/p>\n

      Three reasons why C is the odd one out:<\/strong><\/p>\n

      (A)<\/strong>\u00a0A vertical line fails the vertical line test (a vertical line drawn at x = -2 touches the line at every point). All others pass the vertical line test.
      \n(B)<\/strong>\u00a0C cannot be written as y = mx + b because the slope is undefined. All others can.
      \n(C)<\/strong>\u00a0In C, one input (x = -2) produces infinitely many outputs (any y-value), violating the definition of a function. All others have one output for each input.<\/p>\n

      Conclusion:<\/strong>\u00a0C (vertical line through (-2, 0)) is the odd one out.<\/p>\n

      \u00a0<\/p>\n

      Common Mistakes to Avoid<\/strong><\/p>\n

      Mistake 1 \u2013 Switching rise and run in slope<\/strong>
      \nSlope = rise\/run, not run\/rise.
      \nCorrect understanding: rise is vertical change (y), run is horizontal change (x).<\/p>\n

      Mistake 2 \u2013 Forgetting the negative sign in slope<\/strong>
      \nA negative slope means the line goes DOWN as you move right.
      \nCorrect understanding: Negative rise (go down) with positive run gives negative slope.<\/p>\n

      Mistake 3 \u2013 Thinking y = 3 is not a function<\/strong>
      \nA horizontal line is a function because each x has exactly one y (y = 3 for all x).
      \nCorrect understanding: Only vertical lines are NOT functions.<\/p>\n

      Mistake 4 \u2013 Confusing y-intercept with x-intercept<\/strong>
      \ny-intercept is where x = 0; x-intercept is where y = 0.
      \nCorrect understanding: y-intercept is (0, b); x-intercept is (a, 0).<\/p>\n

      Mistake 5 \u2013 Plotting slope incorrectly from y-intercept<\/strong>
      \nIf slope is 2, from the y-intercept go up 2 AND right 1. Do NOT go up 2 and right 0.
      \nCorrect understanding: Slope 2 = 2\/1, so run must be 1.<\/p>\n

      Mistake 6 \u2013 Believing all straight lines are functions<\/strong>
      \nVertical lines are straight but are NOT functions.
      \nCorrect understanding: A line is a function if it passes the vertical line test.<\/p>\n

      \u00a0<\/p>\n

      Quick Reference Summary<\/strong><\/p>\n

      Linear Function:<\/strong>\u00a0f(x) = mx + b (graph is a straight line)<\/p>\n

      Slope (m):<\/strong>\u00a0m = (y\u2082 – y\u2081) \/ (x\u2082 – x\u2081) = rise\/run<\/p>\n

      Positive slope:<\/strong>\u00a0Line rises from left to right<\/p>\n

      Negative slope:<\/strong>\u00a0Line falls from left to right<\/p>\n

      Zero slope:<\/strong>\u00a0Horizontal line (y = b)<\/p>\n

      Undefined slope:<\/strong>\u00a0Vertical line (x = c) \u2013 NOT a function<\/p>\n

      y-intercept (b):<\/strong>\u00a0Point where line crosses y-axis (0, b)<\/p>\n

      Slope-Intercept Form:<\/strong>\u00a0y = mx + b<\/p>\n

      Standard Form:<\/strong>\u00a0Ax + By = C<\/p>\n

      Point-Slope Form:<\/strong>\u00a0y – y\u2081 = m(x – x\u2081)<\/p>\n

      To Graph a Line:<\/strong>\u00a0Plot y-intercept, use slope to find second point, draw line<\/p>\n

      \u00a0<\/p>\n

      <\/p>\n