Graphing Quadratic Functions

2 Forms of Quadratic Equations y = ax2 + bx + c y = a(x – h)2 + k Standard Form Vertex Form

The axis of symmetry for the parabola is the vertical line through the vertex.

Graphing Using Vertex Form y = a(x – h)2 + k Vertex: (h, k) Axis of symmetry: x = h VERTICAL LINE If a is positive, then it opens up. If a is negative, then it opens down.

Graphing Using Vertex Form Find and sketch the axis of symmetry (opposite of h). Find and plot your vertex (opposite of h, same as k). Construct a table of values to find 2 points on one side of the axis of symmetry (choose 2 x-values above your symmetry value)

Graphing Using Vertex Form Use Symmetry to plot the points on the opposite side of your axis of symmetry. Connect them with a U-shaped curve

f(x)= -3(x – 2)2 + 5 a = -3 h = 2 k = 5 Opens DOWN Tell whether it opens up or down, axis of symmetry, and name the vertex. f(x)= -3(x – 2)2 + 5 y = a(x – h)2 + k. a = -3 h = 2 k = 5 Opens DOWN Axis of symmetry: x = 2 Vertex: (2, 5)

f(x) = (x + 4)2 – 6 k = -6 a = 1 h = -4 Opens UP You try… Tell whether it opens up or down, axis of symmetry, and name the vertex. f(x) = (x + 4)2 – 6 k = -6 a = 1 h = -4 Opens UP Axis of symmetry: x = -4 Vertex: (-4, -6)

OPENS UP Graph h = -5 k = -4 x 2(x + 5)2 - 4 y (x, y) 

Graph h = 3 k= 1 OPENS DOWN x y (x, y) 

Graphing Using Standard Form *Once it is in standard form: Find and sketch the axis of symmetry using Find your vertex by substituting your axis of symmetry back into the original equation and solve for y.

Graphing Using Standard Form 4. Construct a table of values to find 2 points on one side of the axis of symmetry (choose 2 x-values above your symmetry value) 5. Use Symmetry to plot the points on the opposite side of your axis of symmetry. Connect them with a U-shaped curve *Remember: If a is positive, it opens up, if a is negative, it opens down.

OPENS UP Graph a = 1 b = 8 c = 13 x (x)2 + 8x + 13 y (x, y) 

Graph a = -1 b = 2 c = 0 OPENS DOWN x -(x)2 + 2x y (x, y) 

Converting From Vertex Form to Standard Form: y = (x – 3)2 + 5 Step 1: FOIL the binomial Step 2: Multiply the “a” term by what you just foiled Step 3: combine like terms!

Convert the following to standard form: y = 2(x – 4)2 + 6

Convert the following to standard form: y = (x + 3)² + 4

Converting From Standard Form to Vertex Form Step 1: Identify a, b, and c Step 2: find the vertex (h, k) x-coordinate (h) = y-coordinate (k) = substitute the value you found for the x coordinate. Step 3: Substitute a, h, and k into vertex form!

Convert the following to vertex form:  

What is the vertex form of a parabola whose standard form equation is:  

Convert the following to vertex form: