# 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.

## Presentation on theme: "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."— Presentation transcript:

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 x- axis

The x-intercept has a y coordinate of ZERO

To find the x- intercept, 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 1, y 1 ) (x 2, y 2 )

Step 2: Find the RISE between these 2 points Y 2 - Y 1 = 4 - 3 = 1

Step 3: Find the RUN between these 2 points X 2 - X 1 = 5 - 2 = 3

Step 4: Write the RISE over RUN as a ratio Y 2 - Y 1 = 1 X 2 - X 1 3

Where the line crosses the y- axis

The y-intercept has an x- coordinate of ZERO

To find the y- intercept, plug in ZERO for x and solve

y = mx + b m = slope b = y-intercept

Mark a point on the y- intercept

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=ax 2 +bx+c where a0 making a u-shaped graph called a parabola. Example quadratic equation:

The lowest or highest point of a parabola. Vertex Axis of symmetry- The vertical line through the vertex of the parabola. Axis of Symmetry

y=ax 2 + bx + c If a is positive, u opens up If a is negative, u opens down The x-coordinate of the vertex is at To find the y-coordinate of the vertex, plug the x- coordinate into the given eqn. The axis of symmetry is the vertical line x= Choose 2 x-values on either side of the vertex x- coordinate. 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. Vertex: use b=-8 and a=2 Vertex is: (2,-2) Axis of symmetry is the vertical line x=2 Table of values for other points: x yTable of values for other points: x y 06 06 10 10 2-2 2-2 30 30 46 46 * Graph! x=2

(-1,10) (-2,6) (2,10) (3,6) X =.5 (.5,12)

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. Dont 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 -2 3.5 -3 4 -4 3.5 -5 2 Vertex (-3,4) (-4,3.5) (-5,2) (-2,3.5) (-1,2) x=-3

(-1, 11) (0,5) (1,3) (2,5) (3,11) X = 1

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= The x-coordinate of the vertex is 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 To find the y-coord., plug 1 in for x. Vertex (1,9) The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex)The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex) x=1 (-2,0)(4,0) (1,9)

y=2(x-3)(x+1) Open up or down? X-intercepts? Vertex? Axis of symmetry?

(-1,0)(3,0) (1,-8) x=1

The key is to FOIL! (first, outside, inside, last) Ex: y=-(x+4)(x-9)Ex: y=3(x-1) 2 +8 =-(x 2 -9x+4x-36) =3(x-1)(x-1)+8 =-(x 2 -5x-36) =3(x 2 -x-x+1)+8 y=-x 2 +5x+36 =3(x 2 -2x+1)+8 =3x 2 -6x+3+8 y=3x 2 -6x+11

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