Identify Quadratic Functions. Determine characteristics of quadratic functions.

A Quadratic Function is any polynomial of the form: y = ax + bx + c 2

Quadratic Functions have a distinct look called: parabola “Parabola” is a fancy name for “u-shape”.

NO! YES! NO! Linear 3) 1) 2) YES! NO! Absolute Value 5) 4) Determine if the following are quadratic. If it is not, name what type of function it is. NO! YES! NO! Linear 3) 1) 2) YES! NO! Absolute Value 5) 4)

NO! Linear 6) 7) YES! 8) NO! Linear 9) NO! Cubic 10) Determine if the following are quadratic. If it is not, name what type of function it is. NO! Linear 6) 7) YES! 8) NO! Linear 9) NO! Cubic 10) NO! Absolute Value 11) YES!

Every quadratic function will either open UP or DOWN. OPENS UP!! OPENS DOWN!!

If the leading coefficient is POSITIVE the function opens UPWARD. If the leading coefficient is NEGATIVE the function opens DOWNWARD.

Determine if the function opens up or down. 15) Down! 12) Up! 13) Up! Up! 16) 14) Down!

Properties of a parabola: Symmetry All parabolas are: symmetrical. They have a line of symmetry. If we were to fold the parabola in half the line of symmetry would split it exactly in half. Both sides would be exactly the same.

The line of symmetry will always be a vertical line. Therefore, it is written: x = ?

17) 18) 19) 20) Determine the line of symmetry. x = 2.5 x = -1 x = 1

You can also determine the line of symmetry from the function using the following formula: Opposite of b If + then make it – If – then make it +

x = -4 x = 1 21) y = –3x2 + 10x + 9 22) y = 2x2 + x + 3 Determine the line of symmetry. 21) y = –3x2 + 10x + 9 22) y = 2x2 + x + 3 23) y = 0.25x2 + 2x + 3 x = -4 24) y = –3x2 + 6x – 7 x = 1

Properties of a parabola: Vertex: The vertex is like the turning point of the parabola. It is always written (x,y) The vertex will either be a minimum point (smallest point on the graph) or a maximum point (largest point on the graph).

Determine if the following has a MAXIMUM or a MINIMUM. 26) 25) MAX 27) 28) MIN

Determine the vertex. 29) 30) (-3,2) (2,5)

Determine the vertex. 31) (-2,5)

1) Find the x-coordinate using: You can also find the vertex using the formula. Steps: 1) Find the x-coordinate using: 2) Find y by plugging in x to the function. 3) Write as an ordered pair.

(-4,-1) (2,-14) (-2,-4) Determine the vertex. 32) y = 0.25x2 + 2x + 3

Properties of a parabola: Domain – All the x values (horizontal) The domain of a parabola will ALWAYS be all real numbers. Range – All the y values (vertical)

35) 36) 37) D: all real numbers R: y ≥ –4 D: all real numbers R: y ≥ 1