GRAPHING QUADRATIC FUNCTIONS ALGEBRA TWO CHAPTER FIVE QUADRATIC FUNCTIONS SECTION ONE GRAPHING QUADRATIC FUNCTIONS

Graph Quadratic Functions. LEARNING GOALS Graph Quadratic Functions. Use Quadratic Functions to solve real-life problems.

VOCABULARY 1. A quadratic function in standard form is written as: y = ax2 + bx + c , where a is not equal to 0. 2. A parabola is the U-shaped graph of a quadratic function.

VOCABULARY The vertex of a parabola is the lowest point of a parabola that opens up, and the highest point of a parabola that opens down.

VOCABULARY 4. The vertical line passing through the vertex of a parabola, which divides the parabola into two symmetrical parts that are mirror images of each other, is called the axis of symmetry.

THE GRAPH OF A QUADRATIC FUNCTION The graph of y = ax2 + bx + c is a parabola with these characteristics: The parabola opens up if a > 0 and opens down if a < 0. The parabola is wider than the graph of y = x2 if |a| < 1 and narrower than the graph of y = x2 if |a| > 1. The x-coordinate of the vertex is -b/2a. The axis of symmetry is the vertical line x = -b/2a.

Graphing a quadratic formula in the form y=ax2+bx+c Step 1: Identify the coefficients as a, b, c Step 2: Find the x-coordinate of the vertex by using x= -b/2a Step 3: Find the y-coordinate of the vertex by substituting x from step 2 into the original equation and solving for y Step 4: Plot vertex and draw axis of symmetry Step 5: Plot two points on one side of axis of symmetry (pick 2 points for x and find y) Step 6: Use symmetry to plot 2 more points on other side of axis of symmetry Step 7: draw parabola

x = -b/2a = -(-4)/(2(1)) = 2 GRAPHING A QUADRATIC FUNCTION Graph y = x2 - 4x + 3 SOLUTION The coefficients are a = 1, b = -4, and c = 3. Since a > 0, the parabola opens up. To find the x-coordinate of the vertex, substitute 1 for a and -4 for b in the formula x = -b/2a. x = -b/2a = -(-4)/(2(1)) = 2

y = x2 - 4x + 3 y = (2)2 - 4(2) + 3 y = 4 - 8 + 3 y = -1 GRAPHING A QUADRATIC FUNCTION Graph y = x2 - 4x + 3 SOLUTION To find the y-coordinate of the vertex, substitute 2 for x in the original equation, and solve for y. y = x2 - 4x + 3 Write original equation y = (2)2 - 4(2) + 3 Substitute 2 for x. y = 4 - 8 + 3 Multiply out. y = -1 Simplify.

GRAPHING A QUADRATIC FUNCTION Graph y = x2 - 4x + 3 SOLUTION The vertex is (2, -1). Plot two points, such as (1, 0) and (0, 3). Then use symmetry to plot two more points (3, 0) and (4, 3). Draw the parabola. You can confirm that your work is correct by entering the equation into your graphing calculator. Look to see that it matches the graph you made by hand.

GRAPHING A QUADRATIC FUNCTION Practice the technique you just learned by graphing the following: y = -x2+6x-7

GRAPHING A QUADRATIC FUNCTION Practice the technique you just learned by graphing the following: y = -x2+6x-7

VERTEX AND INTERCEPT FORMS FORM OF QUADRATIC FUNCTION CHARACTERISTICS OF GRAPH Vertex form: y = a(x - h)2 + k The vertex is (h, k) The axis of symmetry is x = h. Intercept form: y = a(x - p)(x - q) The x-intercepts are p and q. The axis of symmetry is halfway between (p, 0) and (q, 0). For the forms, the graph opens up if a > 0 and opens down if a < 0.

Step 1: Identify a, h, k Step 2: Identify and plot your vertex (h,k) Step 3: draw your axis of symmetry x=h Step 5: Plot two points on one side of axis of symmetry (pick 2 points for x and find y) Step 6: Use symmetry to plot 2 more points on other side of axis of symmetry Step 7: draw parabola

GRAPHING A QUADRATIC FUNCTION IN VERTEX FORM Graph y = 2(x - 3)2 - 4 SOLUTION Use the form y = a(x - h)2 + k, where a = 2, h = 3, and k = -4. Since a > 0, the parabola opens up. 1. Plot the vertex (h, k) = (3, -4). 2. Plot two points, such as (2, -2) and (1, 4). 3. Use symmetry to plot two more points (4, -2) and (5, 4). 4. Draw the parabola.

Step 1: Identify a, p, q Step 2: Plot the x-intercepts p and q Step 3: Find the axis of symmetry by finding the halfway between (p, 0) and (q,0) Step 4: Find vertex: Plug the x value of the axis of symmetry into the original equation to find the y-coordinate of the vertex. Plot the vertex. Step 5: Draw parabola

GRAPHING A QUADRATIC FUNCTION IN INTERCEPT FORM Graph y = (-1/2)(x - 1)(x + 3) SOLUTION Use the intercept form y = a(x - p)(x - k), where a = (-1/2), p = 1, and q = 3. Since a < 0, the parabola opens down. 1. The x-intercepts are (1, 0) and (-3, 0). 2. The axis of symmetry is x = -1. 3. The x-coordinate of the vertex is -1. The y-coordinate is y = (-1/2)(-1 - 1)(-1 + 3) = 2 4. Draw the parabola.

y = 2(x - 3)(x + 8) y = 2(x2 + 5x - 24) y = 2x2 + 10x - 48 WRITING QUADRATIC FUNCTIONS IN STANDARD FORM Write y = 2(x - 3)(x + 8) in standard form. SOLUTION y = 2(x - 3)(x + 8) Write original equation y = 2(x2 + 5x - 24) Multiply using FOIL. y = 2x2 + 10x - 48 Use Distributive Property.

ASSIGNMENT READ pg. 249-252. WRITE pg. 253-255. #21, #25, #27, #31, #33, #35, #39, #45,# 51

WRITING QUADRATIC FUNCTIONS IN STANDARD FORM Practice the technique you just learned by graphing the following: y = (x + 1)2 - 8 y = -4(x + 2)(x - 2)