Introduction to Quadratics Objectives: Define Quadratic Functions and Parent functions Explore Parameter changes and their effects.

What is a Quadratic Function? A quadratic function is any function whose graph is a parabola. A quadratic equation is any equation that can be written in the form y = ax2 + bx + c. The constants a, b, and c are called the parameters of the equation. These values tell us the shape and location of the parabola.

Parent Functions The parent function is the simplest function of a certain type. It is called this because all of functions in that group look like that and only change location and shape. The linear parent function is y = x. All graphs are straight lines. The quadratic parent function is y = x2. All graphs are parabolas.

Effects on Parameters What happens to the graph of y = ax2 when a is changed? If a > 0, the parabola opens upward. If a < 0, the parabola opens downward.

Examples y = 2x2 opens upward y = -2x2 opens downward

Effects on Parameters If two quadratic functions of the form y = ax2 have different coefficients, then one graph will be wider than the other. Which of these functions produce the widest graph? y = 3x2, y = -5x2, y = ¾ x2

Answer y = 3x2 y = ¾ x2 y = ¾ x2 is wider, so the smaller a is, the wider the graph is. y = -5x2

Graph the following four parabolas on the graph at right and label each A. B. C. D. x y Effect of |a| In a quadratic formula , when |a| increases, the resulting graph is ___________, when |a| decreases, the resulting graph is _____________. narrower wider

Transformations on a graph Reflection is a flip over a line Translation is movement without rotating, resizing or anything else, just moving Every point of the shape must move the same distance in the same direction Rotation means turning around a center **After any of those transformations (turn, flip or slide), the shape still has the same size, area, angles and line lengths

Reflection

Translation

Rotation

Graph the following four parabolas on the graph at right and label each A. B. C. D. x y Effect of –a In a quadratic formula , when a is multiplied by -1 the resulting graph is the same as the original graph _____________________________. Reflected over the x-axis

Effects on Parameters If two quadratic functions of the form y = (x - h)2 have different values for h then one graph will be a translation right or left from the other graph. Compare these three graphs: y = (x – 0)2 y = (x + 1)2 y = (x – 4)2 Parent function! What happens to the parent function when we “add 1”? What happens when we “subtract four”?

Answer y y = (x + 1)2 is translated left one spaces from y = x2 y = (x – 4)2 is translated right four spaces from y = x2 y = x2 - 4 y = (x+1)2

Effects on Parameters If two quadratic functions of the form y = x2 + c have different constants, c, then one graph will be a translation up or down from the other graph. Compare the graphs of y = x2 + 3 with the graph of y = x2 – 4.

Answer y = x2 + 3 y = x2 + 3 is translated up three spaces from y = x2 and y = x2 – 4 is translated down four spaces from y = x2, so there are 7 spaces in between the two graphs. y = x2 - 4

Graph the following four parabolas on the graph at right and label each A. B. C. D. x y Effect of c In a quadratic formula , when c increases one unit, the resulting graph is __________________, when c decreases one unit, the resulting graph is __________________. translated up 1 unit translated down 1 unit

Examples Identify the parent function of the following equations. y = -3x2 + 4x – 7? y = x2 → Quadratic Function 3x – 4y = 8 y = x → Linear Function

Examples - Continued Determine whether the function face upward or downward. y = -3x2 + 2 - Downward y = ½ x2 - Upward

Examples- Continued What is the new equation if the given function is translated down 4 spaces? y = x2 – 3  y = x2 – 7 y = x2 + 8  y = x2 + 4

Examples - Continued Order the equations from widest to most narrow. y = -6x2, y = ¼ x2, y = 4x2 How do the two given equation compare? y = -3x2 – 8 y = x2 y = ¼ x2, y = 4x2, y = -6x2 Translated down 8 spaces, reflected over the x-axis to face downwards and is narrower.

Lesson Check What causes a parabola to move left or right? What causes a parabola to move up or down? What causes a parabola to flip upside down? What causes a parabola to get wider? What causes a parabola to get narrower?