Warm Up Describe the transformation, then graph the function. 1) h(x)= (x + 9)2 + 7 2) g(x) = -5x2 + 0.75 Write the resulting equation for the following transformation of function f: 3) The graph of “g” is a vertical shrink by a factor of 0.25, a reflection over the x – axis followed by a translation 2 units right of f(x) = x2 .

2.2 Notes: Characteristics of Quadratic Functions

Axis of Symmetry Make a t-table for the parent quadratic function from -3 to 3. What do you notice about the y-values?

Parabolic Symmetry As we saw in our t- tables, the y values repeated on either side of the vertex (at the origin for the parent function). This is because parabolas are symmetric, we can draw a line through the vertex and fold the parabola evenly onto itself. This line we draw through the vertex is called the axis of symmetry. It has an equation of x = h (h being the x-value of the vertex, like our vertex form equation). If we know where the vertex is and a few points on either the left or the right, then we can draw the rest of the parabola by mirroring the points.

Let’s Try a Few…. Graph the function, label the vertex and the axis of symmetry: 1) d(x) = -2(x + 3)2 + 4 2) m(x) = 0.5(x – 4)2 – 2

ODDS

Standard Quadratic Function Form Functions can be written in standard quadratic form, which is: f(x) = ax2 + bx + c, where a, b, and c are real numbers. The vertex is not indicated when in standard form. We can find it by: A) x = - B) plug that value in for “x” and find the answer, which will be the y value. “a” is still the reflection, stretch or shrink “c” is the y-intercept. The axis of symmetry is at the x – value of the vertex, or x = -

Let’s Graph Some: Graph the following quadratic functions. Label the vertex and the axis of symmetry. 1) t(x) = 3x2 – 6x + 1 2) r(x) = 0.5x2 – 4x – 2

Closure: Graph the following functions, labeling the vertex and the axis of symmetry: 1) f(x) = -3(x + 1)2 2) g(x) = 2(x – 2)2 + 5 3) h(x) = x2 + 2x – 1 4) p(x) = -2x2 – 8x + 1

ODDS

Warm Up Graph the following quadratic functions, label the vertex and axis of symmetry. 1) v(x) = (x – 3)2 2) w(x) = -4(x – 2)2 + 4 Find the vertex and axis of symmetry: 3) b(x) = -x2 – 1 4) j(x) = 3x2 + 6x – 24 (hint: what’s “b”?)

Maximum and Minimum values A minimum value is the lowest value or point on a parabola (the y value of the vertex). The value will be a minimum when a > 0. When the vertex is a minimum, it is the point where the parabola changes direction from decreasing (on the left of the vertex) to increasing (on the right of the vertex). A maximum value is the highest value or point on a parabola (the y value of the vertex). The value will be a maximum when a < 0. When a vertex is a maximum, it is the point where the parabola changes direction from increasing (on the left of the vertex) to decreasing (on the right of the vertex).

Let’s try some….. Determine if the parabola has a maximum or minimum value, then determine what that value is. 1) q(x) = 0.5x2 – 2x – 1 2) b(x) = -2x2 + 8x – 6

ODDS

Quadratic Intercept Form There are two types of intercepts, y – intercepts (point is on the y – axis) and x – intercepts (point is on the x – axis). Quadratic intercept form gives the x – intercepts of the function. The intercept form is : f(x) = a(x – p)(x – q); where “a” still dictates the direction and steepness of the parabola, and “p” and “q” are the x – intercepts of the parabola. NOTE: Because of the symmetry of parabolas, the vertex should be exactly halfway between the x – intercepts!

Let’s try some: Graph the following functions given in intercept form, label the x – intercepts, maximum or minimum, vertex and axis of symmetry: 1) b(x) = -2(x + 3)(x – 1) 2) d(x) = - (x – 4)(x + 2) 3) f(x) = -(x + 1)(x + 5) 4) g(x) = (x- 6)(x – 2)

p.61-63 #3 – 63 odd

Warm Up 8/14 Find the vertex, AOS, max or min value, and graph the following: 1) d(x) = -(x – 4)2 + 1 2) b(x) = (x + 2)(x – 8)

Linear Modeling On your graphing calculator, we are going to input a list and plot a scatter plot, determine the line of best fit and it’s equation. 1) Femur Length (x) 40 45 32 50 37 41 30 34 47 Height (y) 170 183 151 195 162 174 141 185 182

Now you try one: