Mathematics Dictionary Dr. K. G. Shih

Linear Functions and Equations

---

Subjects

Symbol Defintion Example x^2 = square of x

AL 04 00 | - Outlines AL 04 01 | - Highlight of this lesson AL 04 02 | - What is linear function ? AL 04 03 | - What is linear equation ? AL 04 04 | - Vertical line and horizontal line AL 04 05 | - Study the line y = 2*x + 1 AL 04 06 | - What is the inverse of line y = m*x + n ? AL 04 07 | - Find the inverse of line y = 2*x + 3 ? AL 04 08 | - Find intersection of line y = 2*x + 3 with its inverse ? AL 04 09 | - Find intersection of line y = 2 with its inverse ? AL 04 10 | - Solve abs(x + 2) = 3 where abs means absolute value AL 04 11 | - Graph of y = |x-1| + |x+1| = c AL 04 12 | - Solve |x-1| + |x+1| = 3 AL 04 13 | - Angle between two lines AL 04 14 | - Two lines are parallel AL 04 15 | - Angle between y = 2*x + 3 and its inverse AL 04 16 | - Intersection between y = 2*x + 3 and its inverse AL 04 17 | - Examples

Answers

AL 04 01. Hightlight of this lesson Special topics * Graphic proof of inverse of y = m*x + n is x = m*y + n (Q6) * How to solve abs(m*x + n) = p (See Q09) * Graphic solution of (m*x + n) > p * Graphic solution of (m*x + n) < p How to use computer graphs ? Program ABH - Graphic Calculator Examples * No download and run at current location * Click 01 Linear Function and Equation in upper box * Click program number in lower box * Now it is in graphic mode * Return to menu mode - Click Back command Visit the 130 graphic samples in Program ABI Go to Begin AL 04 02. What is linear function ? Defintion * Linear function : Y = m*x + n Where m is the slope of the line and n is y-intercept The line makes angle A with x-axis and then tan(A) = m If we know slope m then we can find A by arcTan(m) If we know slope m then we can also find A by construction * Linear function is a straight line * Linear function has an inverse function Go to Begin AL 04 03. Slope of line Slope of y = m*x + n Slope is and y-intercept is n m = tan(A) and A is angle of line making with x-axis m = 0, the line is parallel to x-axis m = infinite, the line is vertical A is between 0 and 90, the range is increasing A is between 90 and 180, the range is decreasing Go to Begin AL 04 04. Study the line y = 2 Line y = 2 * It is a horizontal line which cut the y-axis at y = 2. * It is parallel to x-axis and no zero value * Its slope is 0 and y-itercept is 2 * It makes angle A = 0 with line parallel to x-axis * Graphic : Program ABH Topic = 01 and program 01 * Input : m = 0 and n = 2 for y = m*x + n Study the line x = 2 ? * It is a vertical line which cuts the x-axis at x = 2 * It is parallel to y-axis and zero value is 2 * Its slope is infinite and no y-itercept * It makes angle with x-axis A = 90 degree * Graphic : Program ABH Topic = 01 and program 02 * Input : m = 0 and n = 2 for x = m*y + n Go to Begin AL 04 05. Study the line y = 2*x + 1 * It has zero value which is -1/2 * Its slope is 2 * The y-itercept is 1 * It makes angle with x-axis A = arctan(2) degree * Graphic : Program ABH Topic = 01 and program 01 * Input : m = 2 and n = 1 for y = m*x + n Go to Begin AL 04 06. What is the inverse of line y = m*x + n ? * The inverse is the image of y = m*x + n about line y = x * The slope of the inverse is 1/m * The inverse makes angle with x-axis A = arctan(1/m) degree * Property of inverse If y = G(x) is the inverse of y = F(x) The F(G(x)) = x. * Graph of y = m*x + n : Program ABH Topic = 01 and program 01 * Input : m = 2 and n = 1 for y = m*x + n Go to Begin AL 04 07. Find the inverse of line y = 2*x + 3 ? * Graph of y = 2*x + 3 : Program ABH Topic = 01 and program 01 Input : m = 2 and n = 3 for y = m*x + n * Graph of x = 2*y + 3 : Program ABH, Topic = 01 and program 02 Input : m = 2 and n = 3 for x = m*y + n * Graph of x = (y - 3)/2 : Program ABH, Topic = 01 and program 02 Input : m = 0.5 and n = -1.5 for x = m*y + n * Graph of y = (x - 3)/2 : Program ABH, Topic = 01 and program 01 Input : m = 0.5 and n = -1.5 for x = m*y + n * From above examples we see that the inverse is The inverse is y = (x - 3)/2 or The inverse is x = 2*y + 3 Hence the inverse of y = m*x + n is x = m*y + n Go to Begin AL 04 08. Find intersection of line y = 2*x + 3 with its inverse ? * Program ABH Topic = 01 and program 03 * Input : m = 2 and n = 3 for y = m*x + n * Angle of y = 2*x + 3 making with x-axis is A1 = arctan(2) * Angle of inverse making with x-axis is A2 = arctan(1/2) * Angle between line and inverse is A1 - A2 Go to Begin AL 04 09. Find intersection of line y = 2 with its inverse ? * Program ABH Topic = 01 and program 03 * Input : m = 0 and n = 2 for y = m*x + n * Angle of y = 2 making with x-axis is A1 = 90 * Angle of inverse making with x-axis is A2 = 0 * Angle between line and inverse is A1 - A2 = 90 * What is inverse of y = 2 ? Answer : x = 2 Go to Begin AL 04 10. Solve abs(x + 2) = 3 where abs means absolute value * This equation is identical two equations * 1st one is x + 2 = +3 and hence x = +1 * 2nd one is x + 2 = -3 and hence x = -5 * Graphic solution : Intersection of y = abs(x+2) and y = 3 * Program ABH Topic = 01 and program 05 * Input for abs (m*x + n) = p : m = 1, n = 2, p = 3 Go to Begin AL 04 11. Graph of y = abs(x - 1) + abs(x + 1) Graphic is in Program ABH 07 01 This equations contains 3 equations : When x GT 1, y = 2*x. When x between -1 and 1, y = 2. When x LT -1, y = -2*x. Go to Begin AL 04 12. Solve abs(x - 1) + abs(x + 1) = c Solutions are in Program ABH 07 02 Example Solve |x-1| + |x+1| = 3. This equations contains 3 equations : When x GT +1, |x-1| + |x+1| GT 2. When x between -1 and 1, |x-1| + |x+1| = 2. When x LT -1, |x-1| + |x+1| GT 2. Hence there are two roots. When x GT +1, y = 2*x. Hence +2*x = 3 and x = +1.5. When x LT -1, y =-2*x. Hence -2*x = 3 and x = -1.5. Example Solve |x-1| + |x+1| = 1. This equations contains 3 equations : When x GT +1, |x-1| + |x+1| GT 2. When x between -1 and 1, |x-1| + |x+1| = 2. When x LT -1, |x-1| + |x+1| GT 2. Hence there are no solutions. Example Solve |x-1| + |x+1| = 2. This equations contains 3 equations : When x GT +1, |x-1| + |x+1| GT 2. When x between -1 and 1, |x-1| + |x+1| = 2. When x LT -1, |x-1| + |x+1| GT 2. Hence the solution is x between -1 and 1. Go to Begin AL 04 13. Angle between 2 lines Construction Draw line AB and line CD which meets at E. Draw axis Ox and Oy. Definition : Slope of line = tan(U) where U is angle of line making with x-axis. Slope of line AB is m1 = tan(A1). A1 is angle of AB making wiht x-axis. Slope of line CD is m2 = tan(A2). A1 is angle of AB making wiht x-axis. Hence angle between ab and CD is AEC = A1 - A2 Since tan(AEC) = tan(A1-A2) = (tan(A1)-tan(A2))/(1+tan(A1)*tan(A2)) Hence tan(AEC) = (m1 - m2)/(1 + m1*m2) If line AB is perpendicular to CD, then Angle AEC = 90 degrees. tan(90) = infinites. Hence 1 + m1*m2 = 0. This the condition for two line perpemding. Go to Begin AL 04 14. Two lines are parallel Construct a line parallel to other line Construction Draw line AB Draw a line EF which cut AB at P Draw a point Q on line EF Draw angle PQC = 180 - angle QPB QC is the line parallel to AB Condtions for parallel Alternative angles are equal (Angle APQ = Angle PQD). Coresponding angles are equal (Angle EPB = Angle PQD). Angle PQD + angle QPB = pi. Distance between two parallel lines line 1 is y = a*x + b Line 2 is y = a*x + d Distance between two lines is (b - d)*cos(A) Where A is angle of line making with x-axis and A = arctan(a) Go to Begin AL 04 15. Angle between y = 2*x + 3 and its inverse Angle of line y = 2*x + 3 making with x-axis The angle is A1 = arctan(m) where m is the slope and m = 2 Hence A1 = 63.435 degrees Angle of inverse of line y = 2*x + 3 making with x-axis The inverse is x = 2*y + 3 or y = x/2 - 3/2 The angle is A2 = arctan(m) where m is the slope and m = 1/2 Hence A2 = 26.565 degrees Angle between y = 2*x + 3 and its inverse A = A1 - A2 = 63.435 - 26.565 = 36.87 degrees Go to Begin AL 04 16. Intrsection between y = 2*x + 3 and its inverse Answer Find intersection of y = 2*x + 3 and x = 2*y + 3 Hence y = 2*(2*y + 3) + 3 Hence -3*y = 9 and y = -3 Hence x = -3 Intersection point is at (-3, -3) Go to Begin AL 04 17. Examples Study Program | Definition and examples . Study notes of y = a*x + b 1. Linear Functions : It is a line. 2. Slope of line is a. Slope = infinte and A = 90 degrees. It is vertical line. Slope = 0 and A = 0. It is a horizontal line. Slope is positive, it is increasing. Slope is negative, it is decreasing. 3. Angle A of line make with a-axis is A = arctan(Slope). 4. Calculater atn(x) is arctan(x). 5. One degree = pi/180 radians. Study the graph of demo question in Program 01 : What is the expression of the demo function ? What is the y-intercept ? What is the slope of the line ? what is the zero value of y ? What is the angle A of the line making with x-axis ? Verify the estimated value of A using atn(x) on calculator Exercises 1. How to find angles between two lines ? 2. Sketch y = 2. It is zero degree polynomial not linear. 3. sketch y = -2*x +1. Go to Begin AL 04 00. Examples If y = m*x + n Slope = m = tan(A) y-intercept is n The inverse is x = m*y + n Angle between two lines : y = m*x + n and y = p*x + q Angle between two lines = arctan((m - p)/(1 + m*p)) If m = p, then two lines are parallel If 1 + m*p = 0, then two lines are perpendicular Go to Begin

---

Show Room of MD2002

Contact Dr. Shih

Math Examples Room

---

Copyright © Dr. K. G. Shih. Nova Scotia, Canada. URL - www.geocities.com/b192907

---