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6 Linear and Quadratic Functions

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6 Linear and Quadratic Functions 6 Linear and Quadratic Functions 6.1 Prologue Definition 4 in Book I of Euclid’s Elements (3rd century BC) reads: “A straight line is a line that lies evenly with the points on itself.” No one knows for sure what exactly he meant by this — maybe the meaning got lost in translation. Perhaps he meant that if you aim from one point to another, all the points in between fall on that line and no points stick out. In any case, we know what a straight line is. Or do we? A stretched piece of string? A ray of light? This is probably the most deceptive intuitive notion. Einstein’s theory of relativity states that rays of light are actually bent by gravity. In general, our faith in Euclidean geometry has been shaken by more recent (18th century) models of non-Euclidean geometries, in which there exists more than one line through a given point that is parallel to a given line, or in which no parallel lines exist at all. What if our space is actually curved? Here we’ll avoid these mind-bending considerations and stick with the Euclidean point of view. We define a straight line on the Cartesian plane as a graph of a linear relation px+= qy C . If q ≠ 0 , this relation is a function of the form y =+mx b , where p C m =− and b = . A function y = mx+ b is called a linear function. In q q Sections <...>-<...>, we will discuss the properties of linear relations and functions. A function given by the formula f ()xaxbxc= 2 ++, where a, b, and c are constants and a ≠ 0 , is called a quadratic function. Its graph is a parabola, a curve with many 2010 by Skylight Publishing wonderful properties. We will discuss the properties of quadratic functions and their © graphs in Sections <...>-<...>. Copyright 6.2 Linear Relations A relation from \ to \ of the form px+ qy = C , where p, q, and C are constants (and at least one of p and q is not equal to 0) is called a linear relation. The graph of a linear relation in the Cartesian plane is a straight line. 1 2 CHAPTER 6 ~ LINEAR AND QUADRATIC FUNCTIONS It is not very hard to prove that this definition of a straight line is consistent with the postulates of Euclidean geometry. For example, Postulate 1 states that we can draw a straight line through any two points. Indeed, the points (,x y ) and (,x y ) lie on 11 22 the graph of px+= qy C , where p =−()yy21, qxx= −−()21, and Cx= 12y − x 21y (see Question <...> in the exercises). It is also not very hard to prove that there is only one line that can be drawn through any two given points. A straight line is an abstraction: our intuition (or our education) tells us that we can align an imaginary ruler, called a straightedge, with any two points, and draw a straight line through them with an infinitely thin pencil. Example 1 Find a linear relation that defines a straight line through the points (2, 5) and (1, 4). Solution Using the above formulas, p =−()yy21, qxx= −−()21, and Cx= 12y − x 21y , we get: py=−=−=−21 y 45 1, qxx= −−=−−=()(12)121 , Cxyxy= 12−=⋅−⋅= 21 24 15 3. Thus, the relation −+xy =3 or yx− = 3 defines the line through (2, 5) and (1, 4). By our definition, the x-axis is a straight line, because it is the graph of the relation 010⋅+⋅=xy ⇔ y = 0. In general, if p = 0, the graph is the horizontal line C y = . Similarly, the y-axis is a straight line, because it is the graph of q 2010 by Skylight Publishing C © 10⋅+⋅=xy 0 ⇔ x = 0. In general, if q = 0, the graph is the vertical line x = . p Copyright Two straight lines that do not intersect (have no common points) are called parallel. Clearly the graphs of px+ qy = C1 and px+ qy = C2 cannot share a point when C1 and C2 are different, so these lines are parallel. If p and q are fixed, and we vary C, we get a family of parallel lines (Figure 6-1). One of them, the graph of px+= qy 0 , passes through the origin. CHAPTER 6 ~ LINEAR AND QUADRATIC FUNCTIONS 3 y O x pxqyC+ = 1 pxqyC+ = 2 px+ qy = 0 pxqyC+ = 3 Figure 6-1. The graphs of px + qy = C for different values of C produce a family of parallel lines This model of Euclidean geometry is also consistent with Euclid’s fifth postulate, which is equivalent to the statement that given a straight line l and a point P outside it, we can draw precisely one line through P parallel to l: l P 2010 by Skylight Publishing © If a straight line is neither vertical nor horizontal, that is, p ≠ 0 and q ≠ 0 , and it Copyright doesn’t pass through the origin, that is C ≠ 0 , then it has one x-intercept, a ≠ 0 and one y-intercept, b ≠ 0 : y b O x a 4 CHAPTER 6 ~ LINEAR AND QUADRATIC FUNCTIONS xy 1 1 An equation of such a line can be written as + =1, where p = , q = , and ab a b C =1). This form of the equation of the line is called the intercept form. Example 2 Sketch a straight line through (2, 0) and (0, 1) and determine its equation. Solution y 1 -2 -1 O 1 2 x -1 The x-intercept is 2 and the y-intercept is 1. Using the intercept form, we xy immediately find the equation: + =1 or xy+ 22= . 21 Example 3 2010 by Skylight Publishing © Sketch the graph of x + 2y = 3. Copyright Solution We know from Example 2 what the graph of x + 2y = 2 looks like. The graph x + 2y = 3 is a line parallel to the line x + 2y = 2, but its x-intercept is 3, not 2: CHAPTER 6 ~ LINEAR AND QUADRATIC FUNCTIONS 5 y 1 -1 O 1 2 3 x -1 To solve this problem “from scratch,” set y to 0 to get the x-intercept, 3; then set x to 3 0 to get the y-intercept, , then connect the two intercept points with the straight 2 line. Any line px+ qy = C1 is perpendicular to any line ()−qx+= py C2 . Recall from Chapter <....> that two vectors are perpendicular to each other if and ± G only if their dot product is equal to 0. Also recall that the vector v , which represents G the difference of the vectors (,x22y ) and (,x11y ), vxx= (,2121−−yy ), is parallel to the line that passes through the points (,x11y ) and (,x22y ): y (,xy ) 11G v (,xy22 ) O x G Suppose an equation of this line is px+ qy = C . If upq= (,), then the dot product 2010 by Skylight Publishing GG G © uv⋅= px()() − x + qy − y = ()()px+ qy−+=−= px qy C C 0. This means u 21G 21 22 11 is perpendicular to v . Thus a line px+ qy = C is perpendicular to the vector G upq= (,). We can see it graphically, too. For example: Copyright y 2 (1, 2) 1 x + 2y = 3 -1 O 1 2 3 x -1 6 CHAPTER 6 ~ LINEAR AND QUADRATIC FUNCTIONS G G If we rotate u by 90 degrees, we get the vector uqp= (,)− : it has the same length G ⊥ G G as u and is perpendicular to it. Indeed, their dot product, uu⋅ ⊥ =−+= p() q qp 0. ¯ Example 4 Find an equation of the line that contains the point (1,3)− and is perpendicular to the line 351x +=y . Solution The equation has the form (5)− x += 3y C . For the point (1,3)− to be on that line, we must have (5)(1)33−−+⋅=CC ⇒ =14. The answer is −5314xy+=. Exercises 1. When and where did Euclid live? How many books make up Euclid’s Elements? In Questions 2-5, write an equation px+= qy C for the straight line that passes through the two given points. 2. (-2, -1) and (3, 5) 3 3. (2, 4) and (-2, -2) 4. (2, 2) and (2, 5) 3 2010 by Skylight Publishing © 5. (1, -2) and (-3, -2) In Questions 6-9, write an equation for the straight line that passes through the given Copyright point and is parallel to the given line. 6. Point (-3, 1) and line xy+=0 . 3 7. Point (2, 5) and line y = 0 . 8. Point (4, -1) and line xy+=37. 3 CHAPTER 6 ~ LINEAR AND QUADRATIC FUNCTIONS 7 9. Point (-1, 2) and line 22xy− = . In Questions 10-13, write an equation for the straight line through the two given xy points in the intercept form, +=1. ab 10. (0, 3) and (4, 0). 3 11. (0, -2) and (3, 0). 12. (4, 0) and (0, -1). 3 13. (-3, 2) and (2, -3). ÒHint: First write px+ qy = C , then find the intercepts.Ñ In Questions 14-17, write an equation for the straight line that passes through the given point and is perpendicular to the given line. 14. Point (0, 0) and line x +=y 1. 3 15. Point (6, -3) and line x = 2 . 16. Point (3, -3) and line xy−=0 . 3 17. Point (-1, 2) and line 22xy− = . 18. Prove that the line px+= qy C , where p = ()yy21− , qxx= −−()21, and Cx= 12y − x 21y contains the points (,x11y ) and (,x22y ). 2010 by Skylight Publishing © 19. Prove that only one straight line passes through the points (,x11y ) and (,x22y ).
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