Graphing Quadratic Functions (10.1) Objective: Students will sketch the graph of a quadratic function.

Algebra Standards: 16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. 21.0 Students graph quadratic functions and know that their roots are the x-intercepts. 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity.

Vocabulary Quadratic Function: is a function that can be written in the Standard form y = ax2 + bx + c, where a ≠ 0 Parabola: is the U-shaped graph of a quadratic Function.

is the lowest point on the graph of a parabola Vocabulary Vertex: is the lowest point on the graph of a parabola opening up or the highest point on the graph of parabola opening down. is the vertical line passing through the vertex of a parabola and divides the parabola into two symmetrical parts that mirror images of each other. Axis of Symmetry

-a -c Vocabulary Quadratic Equation Quadratic Term Linear Term Constant Term a c +a opens up y-intercept -a opens down +c shifts up -c skinny parabola shifts down wide parabola

Graphing a Quadratic Function To graph y = ax2 + bx + c, where a ≠ 0, is a parabola. Step 1: Determine the values of a, b, and c from the equation Step 2: Determine if it opens up (+a) or down up(-a). b – Step 3: Find the Axis of Symmetry, x = 2a Step 4: Make a t-table, using x-values to the left and right of the Axis of Symmetry Step 5. Plot the Points

b – – – 2a 2(1) 2 Step 1: a = 1, b = 0, c = 0 Step 2: #1 Graph a quadratic Function Sketch the graph of y = x2 Step 1: a = 1, b = 0, c = 0 Step 2: Since a = +1, therefore it opens up. b – – – Step 3: A . S = = = = 2a 2(1) 2 Step 4: x y = x2 y (x, y) -3 (-3)2 9 (-3, 9) -2 (-2)2 4 (-2, 4) Vertex = (0, 0) -1 (-1)2 1 (-1, 1) 02 (0, 0) Axis of Symmetry x = 0 1 12 1 (1, 1) 2 22 4 (2, 4) 3 32 9 (3, 9)

Step 5: Vertex = (0, 0) Opens up Axis of Symmetry x = 0 (x, y) #1 Graph a quadratic Function Sketch the graph of y = x2 Step 5: Opens up x y Vertex = (0, 0) Axis of Symmetry x = 0 (x, y) (-3, 9) (-2, 4) (-1, 1) (0, 0) (1, 1) (2, 4) (3, 9)

b – – – 2a 2(1) 2 Step 1: a = 1, b = 0, c = 2 Step 2: #2 Graph a quadratic Function Sketch the graph of y = x2 + 2 Step 1: a = 1, b = 0, c = 2 Step 2: Since a = +1, therefore it opens up. b – – – Step 3: A. S (x)= = = = 2a 2(1) 2 Step 4: x y = x2+ 2 y (x, y) -3 (-3)2 + 2 11 (-3, 11) -2 (-2)2 + 2 6 (-2, 6) Vertex = (0, 2) -1 (-1)2 + 2 3 (-1, 3) 02 + 2 2 (0, 2) Axis of Symmetry x = 0 1 12 + 2 3 (1, 3) 2 22 + 2 6 (2, 6) 3 32 + 2 11 (3, 11)

Step 5: Vertex = (0, 2) Opens up Axis of Symmetry x = 0 (x, y) #2 Graph a quadratic Function Sketch the graph of y = x2 + 2 Step 5: Opens up x y Vertex = (0, 2) Axis of Symmetry x = 0 (x, y) (-3, 11) (-2, 6) (-1, 3) (0, 2) (1, 3) (2, 6) (3, 11)

Vertex = (0, -4) Opens up Axis of Symmetry x = 0 #3 Graph a quadratic Function Sketch the graph of y = x2 – 4 x y Opens up Vertex = (0, -4) Axis of Symmetry x = 0

b 4 4 – – – 1 2a 2(-2) -4 Step 1: a = -2, b = 4, c = 1 Step 2: #4 Graph a quadratic Function Sketch the graph of y = -2x2 + 4x + 1 Step 1: a = -2, b = 4, c = 1 Step 2: Since a = -2, therefore it opens down. b 4 4 – – – Step 3: A.S (x)= = = = 1 2a 2(-2) -4

Step 4: x y = -2x2 + 4x + 1 y (x, y) -2 -1 Vertex = (1, 3) 1 2 3 4 #4 Graph a quadratic Function Step 4: x y = -2x2 + 4x + 1 y (x, y) -2 -2• (-2)2 + 4(-2) +1 -15 (-2, -15) -8 – 8 + 1 -2• (-1)2 + 4(-1) +1 -5 -1 (-1, -5) -2 – 4 + 1 -2• (0)2 + 4(0) + 1 1 (0, 1) + 0 + 1 Vertex = (1, 3) 1 -2• (1)2 + 4(1) +1 3 (1, 3) -2 + 4 + 1 Axis of Symmetry x = 1 2 1 (2, 1) 3 -5 (3, -5) 4 -15 (4, -15)

Step 5: Vertex = (1, 3) Opens down Axis of Symmetry x = 1 (x, y) #4 Graph a quadratic Function Sketch the graph of y = -2x2 + 4x + 1 Step 5: Opens down x y Vertex = (1, 3) Axis of Symmetry x = 1 (x, y) (-2, -15) (-1, -5) (0, 1) (1, 3) (2, 1) (3, -5) (4, -15)

Finding Critical Features of Quadratics Vertex Axis of Symmetry Opens Up/Down Opens Up Y-intercept (0, 2)

Find the critical features of the quadratic below. Vertex Axis of Symmetry Opens Up/Down Opens Down Y-intercept (0, 0)

Find the critical features of the quadratic below. Vertex Axis of Symmetry Opens Up/Down Opens Up Y-intercept (0, 4)