{"id":2599,"date":"2023-10-03T10:31:54","date_gmt":"2023-10-03T10:31:54","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=2599"},"modified":"2023-10-03T10:47:41","modified_gmt":"2023-10-03T10:47:41","slug":"week-3-ss1-third-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-3-ss1-third-term-further-mathematics-notes\/","title":{"rendered":"Week 3 – SS1 Third Term Further Mathematics Notes"},"content":{"rendered":"

WEEK 3
\n<\/strong>Topic:<\/em><\/strong> \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Gradients of straight lines and curves
\nSub-topic:<\/em>
\n\t\t\t\t<\/strong>\u00a0\u00a0\u00a0\u00a0Gradients of straight lines
\nDuration:<\/em><\/strong> 40 minutes
\nLearning Objectives:<\/em><\/strong> By the end of the lesson, students should be able to calculate the gradient of a straight line.
\nReference Materials:<\/em><\/strong> i. New General Mathematics for SSS 2, by M.F Macrae et al. Pages 184 \u2013 192.
\nPrevious Knowledge<\/em>:<\/strong> Students can draw the graph of a linear equation (straight-line graph).
\nInstructional Materials<\/em>:<\/strong> Graph board and graph book.
\nContent:\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/em>GRADIENT OF A STRAIGHT LINE
\n\t\t\t\t\t<\/em><\/strong>The gradient of a straight line is the rate of change of y compared with x.
\nFor example, if the gradient is 2, then for any increase in x, y increases two times as much.
\n\"\"\/<\/p>\n

\u00a0
\n\u00a0
\n\u00a0
\n\u00a0Gradient of AB =\u00a0\u00a0\u00a0\u00a0Increase in y from A to B = MB\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Increase in x from A to B\u00a0\u00a0\u00a0\u00a0AM<\/p>\n

\u00a0Example
\n<\/strong>Find the gradient of the line joining P(7, -2) and Q(-1, 2)
\n\"\"\/\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\n

\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0Gradient of PQ = increase in y = – AQ<\/p>\n

\t\t\t\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 Increase in x PA
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 =
\n\t\t\t\t<\/strong>
\n\u00a0Example 2
\n<\/strong>Find the gradient of the line 7x + 4y \u2013 8 = 0
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Re-arrange the equation: 4y = – 7x + 8
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 y = + 2
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Therefore, gradient (m) = , y \u2013 intercept (c) = 2
\nSKETCHING GRAPHS OF STRAIGHT LINES
\n\t\t\t\t\t<\/em><\/strong>\u00a0\u00a0\u00a0\u00a0Given the equation
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0y = 3x \u2013 2 , gradient = 3, y \u2013 intercept(c) = -2
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a02x + 3y = 6, gradient = , y \u2013 intercept(c) = 2<\/p>\n

\u00a0Example
\n<\/strong>Sketch the graph of the line whose equation is 4x \u2013 3y = 12
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Solution<\/strong>
\n\t\tWhen x = 0 ,- 3y = 12
\n y = – 4
\nThe line crosses the y \u2013 axis at (0, – 4).
\nWhen y = 0 , 4x = 12
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 x = 3
\nThe line crosses the x \u2013 axis at (3, 0).
\n\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0From the graph:
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Gradient m = <\/p>\n

\u00a0 = =
\n y \u2013 intercept = – 4<\/p>\n

\u00a0
\n\u00a0
\n\u00a0Lines parallel to axes
\n<\/strong>Any line parallel to the x \u2013 axis has a gradient of zero. The equation of such lines is always in the form
\ny = c<\/em><\/strong>, where c<\/em><\/strong> may be any number.
\nThe figure below shows the graph of y = 5 and y = – 3.
\nNotice that the equation of the x \u2013 axis is y = 0
\n\"\"\/
\n\t\t\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0————————————<\/p>\n

\u00a0
\n\u00a0
\n\u00a0
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0————————————-<\/p>\n

\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0The gradient of a line that is parallel to the y \u2013 axis is undefined. The equations of such a lines are always in the form x = a<\/em><\/strong> , where a<\/em><\/strong> may be any number.
\nThe figure below shows the graph of line x = 2 and x = – 4.
\nNotice that the equation of the y \u2013 axis is x = 0<\/p>\n

\u00a0\"\"\/
\n\t\t\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\n

\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0
\n\u00a0EQUATION OF A STRAIGHT LINE
\n\t\t\t\t\t<\/em><\/strong>Equation of a straight line is of the form y = mx + c, where m is the gradient and c is the y \u2013 intercept.<\/p>\n

\u00a0Example 1
\n<\/strong>Determine the equation of a straight line whose gradient is and passes through the point (- 3, 2).
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Solution<\/strong>
\n\t\tUsing the formula y \u2013 y1<\/sub> = m(x – x1<\/sub>)
\nWhere (x1<\/sub>, y1<\/sub>) = (- 3, 2) and m =
\n\u00a0\u00a0\u00a0\u00a0y \u2013 2 = (x + 3)
\n\u00a0\u00a0\u00a0\u00a03y \u2013 6 = – x \u2013 3
\n\u00a0\u00a0\u00a0\u00a0x + 3y = 3<\/p>\n

\u00a0Example 2
\n<\/strong>Find the equation of the straight line passing through the points (1, 4) and (- 2, 6).
\nUsing the formula
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 =
\nWhere = (x1<\/sub>, y1<\/sub>) = ( 1, 4) and (x2<\/sub>, y2<\/sub>) = (- 2, 6), the equation is
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 =
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0cross multiply
\n \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0– 3y + 12 = 2x \u2013 2
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 2x + 3y = 14
\nGRADIENT OF A CURVE
\n\t\t\t\t\t<\/em><\/strong>Example
\n<\/strong>Draw the graph of y = for values of x from \u20132 to 3. Find the gradient of the curve at the point where x has the value (a) (b) \u2013 2
\n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Solution
\n<\/strong><\/p>\n

\n\n\n\n\n
x<\/td>\n-2<\/td>\n-1<\/td>\n0<\/td>\n1<\/td>\n2<\/td>\n3<\/td>\n<\/tr>\n
y<\/td>\n1<\/td>\n\u00bc<\/td>\n0<\/td>\n\u00bc<\/td>\n1<\/td>\n2\u00bc<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

\u00a0(a) Gradient of the curve where x = 3
\n\u00a0\u00a0\u00a0\u00a0= gradient of tangent PT
\n\u00a0\u00a0\u00a0\u00a0= \u00a0\u00a0\u00a0\u00a0= 2.25\u00a0\u00a0\u00a0\u00a0 = 2\u00bc = 9 = 11<\/sup>\/2<\/sub>\u00a0\u00a0\u00a0\u00a0
\n\u00a0\u00a0\u00a0\u00a0 1.5\u00a0\u00a0\u00a0\u00a0 1.5 6
\n(b) Gradient of curve where x = – 2
\n\u00a0\u00a0\u00a0\u00a0= gradient of tangent QR
\n = – = – 1\u00a0\u00a0\u00a0\u00a0= – 1 \u00a0\u00a0\u00a0\u00a0
\n\u00a0\u00a0\u00a0\u00a0 1<\/p>\n","protected":false},"excerpt":{"rendered":"

WEEK 3 Topic: \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Gradients of straight lines and curves Sub-topic: \u00a0\u00a0\u00a0\u00a0Gradients of straight lines Duration:…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,216],"tags":[],"class_list":["post-2599","post","type-post","status-publish","format-standard","hentry","category-posts","category-third-term-ss1-further-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2599","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/comments?post=2599"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2599\/revisions"}],"predecessor-version":[{"id":2600,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2599\/revisions\/2600"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=2599"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=2599"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=2599"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}