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An equation for which the graph is a line Any ordered pair of numbers that makes a linear equation true. (9,0) IS ONE SOLUTION FOR Y = X - 9 Example: y=x+3 Step 1: ~ Three Point Method ~ Choose 3 values for x Step 2: Find solutions using table y=x+3 Y | X 0 1 2 Step 3: Graph the points from the table (0,3) (1,4) (2,5) Step 4: Draw a line to connect them Graph using a table (3 point method) 1) y = x + 3 2) y = x - 4 Where the line crosses the xaxis The x-intercept has a y coordinate of ZERO To find the xintercept, plug in ZERO for y and solve Describes the steepness of a line Equal to: Rise Run The change vertically, the change in y The change horizontally or the change in x Step 1: Find 2 points on a line (2, 3) (5, 4) (x , y ) (x , y ) 1 1 2 2 Step 2: Find the RISE between these 2 points Y-Y = 4-3=1 2 1 Step 3: Find the RUN between these 2 points X-X = 5-2=3 2 1 Step 4: Write the RISE over RUN as a ratio Y-Y 2 1 X-X 2 1 = 1 3 Where the line crosses the yaxis The y-intercept has an xcoordinate of ZERO To find the yintercept, plug in ZERO for x and solve y = mx + b m = slope b = y-intercept Mark a point on the yintercept Define slope as a fraction... (RISE) Denominator is the horizontal change (RUN) Graph at least 3 points and connect the dots Definitions 3 forms for a quad. function Steps for graphing each form Examples Changing between eqn. forms A function of the form y=ax2+bx+c where a≠0 making a u-shaped graph called a parabola. Example quadratic equation: The lowest or highest point of a parabola. Vertex Axis of symmetry Axis of Symmetry The vertical line through the vertex of the parabola. y=ax2 + bx + c If a is positive, u opens up If a is negative, u opens down b The x-coordinate of the vertex is at 2 a To find the y-coordinate of the vertex, plug the xcoordinate into the given eqn. The axis of symmetry is the vertical line x= Choose 2 x-values on either side of the vertex xcoordinate. Use the eqn to find the corresponding y-values. Graph and label the 5 points and axis of symmetry on a coordinate plane. Connect the points with a smooth curve. a=2 Since a is positive the parabola will open up. b Vertex: use x 2a b=-8 and a=2 (8) 8 x 2 2(2) 4 y 2(2) 2 8(2) 6 y 8 16 6 2 Vertex is: (2,-2) • Axis of symmetry is the vertical line x=2 •Table of values for other points: x y 0 6 1 0 2 -2 3 0 4 6 * Graph! x=2 (.5,12) (-1,10) (2,10) (-2,6) (3,6) X = .5 y=a(x-h)2+k If a is positive, parabola opens up If a is negative, parabola opens down. The vertex is the point (h,k). The axis of symmetry is the vertical line x=h. Don’t forget about 2 points on either side of the vertex! (5 points total!) y=2(x-1)2+3 Open up or down? Vertex? Axis of symmetry? Table of values with 5 points? a is negative (a = -.5), so parabola opens down. Vertex is (h,k) or (-3,4) Axis of symmetry is the vertical line x = -3 Table of values x y -1 2 Vertex (-3,4) -2 3.5 (-4,3.5) (-2,3.5) -3 4 -4 3.5 (-5,2) (-1,2) -5 2 x=-3 (-1, 11) (3,11) X=1 (0,5) (2,5) (1,3) y=a(x-p)(x-q) The x-intercepts are the points (p,0) and (q,0). The axis of symmetry is the vertical line x= p q 2 pq The x-coordinate of the vertex is 2 To find the y-coordinate of the vertex, plug the x-coord. into the equation and solve for y. If a is positive, parabola opens up If a is negative, parabola opens down. Since a is negative, parabola opens down. The x-intercepts are (-2,0) and (4,0) To find the x-coord. of the vertex, use p q 24 2 x 1 2 2 •The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex) (1,9) 2 To find the y-coord., plug 1 in for x. (-2,0) (4,0) y (1 2)(1 4) (3)(3) 9 Vertex (1,9) x=1 y=2(x-3)(x+1) Open up or down? X-intercepts? Vertex? Axis of symmetry? x=1 (-1,0) (3,0) (1,-8) The key is to FOIL! (first, outside, inside, last) Ex: y=-(x+4)(x-9) Ex: y=3(x-1)2+8 =-(x2-9x+4x-36) =3(x-1)(x-1)+8 =-(x2-5x-36) =3(x2-x-x+1)+8 y=-x2+5x+36 =3(x2-2x+1)+8 =3x2-6x+3+8 y=3x2-6x+11