Quadratic Functions The graph of a quadratic function is a parabola . A parabola can open up or down. If the parabola opens up, the lowest point is called the vertex. If the parabola opens down, the vertex is the highest point. y x Vertex Vertex

Standard Form:

y = ax 2 + bx + c The parabola will open down when the a value is negative. The parabola will open up when the a value is positive. Standard Form y x The standard form of a quadratic function is a > 0 a < 0

Line of Symmetry:

y x Line of Symmetry Line of Symmetry Parabolas have a symmetric property to them. If we drew a line down the middle of the parabola, we could fold the parabola in half. We call this line the line of symmetry . The line of symmetry ALWAYS passes through the vertex. Or, if we graphed one side of the parabola, we could “fold” (or REFLECT ) it over, the line of symmetry to graph the other side.

Finding the Line of Symmetry:

Find the line of symmetry of y = 3 x 2 – 18 x + 7 Finding the Line of Symmetry When a quadratic function is in standard form The equation of the line of symmetry is y = ax 2 + bx + c , For example… Using the formula… This is best read as … the opposite of b divided by the quantity of 2 times a . Thus, the line of symmetry is x = 3.

Finding the Vertex:

Finding the Vertex The line of symmetry gives us the x – coordinate of the vertex. To find the y – coordinate of the vertex, we need to plug the x – value into the original equation. STEP 1: Find the line of symmetry STEP 2: Plug the x – value into the original equation to find the y value. y = –2 x 2 + 8 x –3 y = –2(2) 2 + 8(2) –3 y = –2(4)+ 8(2) –3 y = –8+ 16 –3 y = 5 Therefore, the vertex is (2 , 5)

A Quadratic Function in Standard Form:

A Quadratic Function in Standard Form STEP 1 : Find the line of symmetry STEP 2 : Find the vertex STEP 3 : Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. Plug in the line of symmetry ( x – value) to obtain the y – value of the vertex. USE the equation MAKE A TABLE using x – values close to the line of symmetry. There are 3 steps to graphing a parabola in standard form.

Slide 8:

STEP 1 : Find the line of symmetry Let's graph one! Try … y = 2 x 2 – 4 x – 1 a = 2 b = -4 A Quadratic Function in Standard Form y x Thus the line of symmetry is x = 1

Slide 9:

STEP 2 : Find the vertex A Quadratic Function in Standard Form y x Thus the vertex is (1 ,–3). We need to plug in x = 1 to find the y – value of the vertex. We know the line of symmetry always goes through the vertex.

Slide 10:

STEP 3 : Find two other points and reflect them across the line of symmetry. Then connect the five points with a smooth curve. A Quadratic Function in Standard Form y x 5 3 -1 2 y x

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