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Graphing Linear Equations. Linear Equation. An equation for which the graph is a line. Solution. Any ordered pair of numbers that makes a linear equation true. (9,0) IS ONE SOLUTION FOR Y = X - 9. Linear Equation. Example: y = x + 3. Graphing. Step 1: ~ Three Point Method ~

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## Graphing Linear Equations

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**Linear Equation**An equation for which the graph is a line**Solution**Any ordered pair of numbers that makes a linear equation true. (9,0) IS ONE SOLUTION FOR Y = X - 9**Linear Equation**Example: y = x + 3**Graphing**Step 1: ~ Three Point Method ~ Choose 3 values for x**Graphing**Step 2: Find solutions using table y = x + 3 Y | X 0 1 2**Graphing**Step 3: Graph the points from the table (0,3) (1,4) (2,5)**Graphing**Step 4: Draw a line to connect them**Try These**• Graph using a table (3 point method) 1) y = x + 3 2) y = x - 4**X-intercept**Where the line crosses the x-axis**X-intercept**The x-intercept has a y coordinate of ZERO**X-intercept**To find the x-intercept, plug in ZERO for y and solve**Slope**Describes the steepness of a line**Slope**Equal to: Rise Run**Rise**The change vertically, the change in y**Run**The change horizontally or the change in x**Finding Slope**Step 1: Find 2 points on a line (2, 3) (5, 4) (x1, y1) (x2, y2)**Finding Slope**Step 2: Find the RISE between these 2 points Y2 - Y1 = 4 - 3 = 1**Finding Slope**Step 3: Find the RUN between these 2 points X2 - X1 = 5 - 2 = 3**Finding Slope**Step 4: Write the RISE over RUN as a ratio Y2 - Y1= 1 X2 - X1 3**Y-intercept**Where the line crosses the y-axis**Y-intercept**The y-intercept has an x-coordinate of ZERO**Y-intercept**To find the y-intercept, plug in ZERO for x and solve**Slope-Intercept**y = mx + b m = slope b = y-intercept**Step 1:**Mark a point on the y-intercept**Step 2:**Define slope as a fraction...**Step 3: Numerator is the vertical**change (RISE)**Step 4:**Denominator is the horizontal change (RUN)**Step 5:**Graph at least 3 points and connect the dots**Graphing Quadratic Functions**• Definitions • 3 forms for a quad. function • Steps for graphing each form • Examples • Changing between eqn. forms**Quadratic Function**• A function of the form y=ax2+bx+c where a≠0 making a u-shaped graph called a parabola. Example quadratic equation:**Vertex-**• The lowest or highest point of a parabola. Vertex Axis of symmetry- • The vertical line through the vertex of the parabola. Axis of Symmetry**Standard Form Equation**y=ax2 + 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.**Example 1: Graph y=2x2-8x+6**• 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! • a=2 Since a is positive the parabola will open up. • Vertex: use b=-8 and a=2 Vertex is: (2,-2) x=2**Now you try one!y=-x2+x+12* Open up or down?* Vertex?***Axis of symmetry?* Table of values with 5 points?**(.5,12)**(-1,10) (2,10) (-2,6) (3,6) X = .5**Vertex Form Equation**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!)**Now you try one!**y=2(x-1)2+3 • Open up or down? • Vertex? • Axis of symmetry? • Table of values with 5 points?**Example 2: Graphy=-.5(x+3)2+4**• 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) (-2,3.5) (-5,2) (-1,2) x=-3**(-1, 11)**(3,11) X = 1 (0,5) (2,5) (1,3)**Intercept Form Equation**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.**Example 3: Graph y=-(x+2)(x-4)**• The axis of symmetry is the vertical line x=1 (from the x-coord. of the vertex) • 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) (1,9) (-2,0) (4,0) x=1**Now you try one!**y=2(x-3)(x+1) • Open up or down? • X-intercepts? • Vertex? • Axis of symmetry?**x=1**(-1,0) (3,0) (1,-8)**Changing from vertex or intercepts form to standard form**• 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

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