Graphing Quadratic Functions

Graphing Quadratic Functions MA.912.A.7.1 Graph quadratic equations. MA.912.A.7.6 Identify the axis of symmetry, vertex, domain, range, and intercept(s) for a given parabola

Quadratic Function y = ax2 + bx + c Quadratic Term Linear Term Constant Term What is the linear term of y = 4x2 – 3? 0x What is the linear term ofy = x2- 5x? -5x What is the constant term of y = x2 – 5x? 0 Can the quadratic term be zero? No!

y Vertex x Vertex Quadratic Functions parabola The graph of a quadratic function is a: A parabola can open up or down. If the parabola opens up, the lowest point is called the vertex (minimum). If the parabola opens down, the vertex is the highest point (maximum). NOTE: if the parabola opens left or right it is not a function!

y a > 0 x a < 0 Standard Form The standard form of a quadratic function is: y = ax2 + bx + c The parabola will open up when the a value is positive. The parabola will open down when the a value is negative.

Axis of Symmetry y x Axis of Symmetry Parabolas are symmetric. If we drew a line down the middle of the parabola, we could fold the parabola in half. We call this line the Axis of symmetry. If we graph one side of the parabola, we could REFLECT it over the Axis of symmetry to graph the other side. The Axis of symmetry ALWAYS passes through the vertex.

Finding the Axis of Symmetry When a quadratic function is in standard form y = ax2 + bx + c, the equation of the Axis of symmetry is This is best read as … ‘the opposite of b divided by the quantity of 2 times a.’ Find the Axis of symmetry for y = 3x2 – 18x + 7 The Axis of symmetry is x = 3. a = 3 b = -18

The Axis of symmetry always goes through the _______. Thus, the Axis of symmetry gives us the ____________ of the vertex. Finding the Vertex Vertex X-coordinate Find the vertex of y = -2x2 + 8x - 3 STEP 1: Find the Axis of symmetry The x-coordinate of the vertex is 2 a = -2 b = 8

Finding the Vertex Find the vertex of y = -2x2 + 8x - 3 STEP 1: Find the Axis of symmetry STEP 2: Substitute the x – value into the original equation to find the y –coordinate of the vertex. The vertex is (2 , 5)

Graphing a Quadratic Function There are 3 steps to graphing a parabola in standard form. STEP 1: Find the Axis of symmetry using: STEP 2: Find the vertex STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. MAKE A TABLE using x – values close to the Axis of symmetry.

y x Graphing a Quadratic Function STEP 1: Find the Axis of symmetry STEP 2: Find the vertex Substitute in x = 1 to find the y – value of the vertex.

y x y x 2 3 Graphing a Quadratic Function STEP 3: Find two other points and reflect them across the Axis of symmetry. Then connect the five points with a smooth curve. –1 5

Y-axis y x Y-intercept of a Quadratic Function The y-intercept of a Quadratic function can Be found when x = 0. The constant term is always the y- intercept

Solving a Quadratic The x-intercepts (when y = 0)of a quadratic function are the solutions to the related quadratic equation. The number of real solutions is at most two. One solution X = 3 Two solutions X= -2 or X = 2 No solutions

Identifying Solutions Find the solutions of 2x - x2 = 0 The solutions of this quadratic equation can be found by looking at the graph of f(x) = 2x – x2 The x-intercepts(or Zero’s) of f(x)= 2x – x2 are the solutions to 2x - x2 = 0 X = 0 or X = 2

Graphing Quadratic Functions