Honors Algebra 2- Unit 2B Practice and Definitions

What is the discriminant expression?

b^2-4ac

How would you find the x-intercepts, AOS, and vertex from factored form? (3 steps)

1. Find the x-intercepts by solving for 0 2. Find the AOS by averaging the x-intercepts 3. Plug the AOS into the original function to find the vertex Use these to solve

What is the order of transformations?

1. Horizontal Translation 2. Reflection/vertical stretch/shrink 3. Vertical translation

What are the steps to converting the standard form quadratic equation to vertex form? (6 steps)

1. Move c to the other side 2. Factor out "a" from right side 3. Complete the square for the expression inside the (). 4. To balance the equation, we multiply the value from step 3 to "a" and add to the left side. 5. Factor the quadratic in (). 6. Isolate f(x).

For each function, identify all transformations of the function f(x) = x^2 Then graph the function. 1. g(x) = x^2 + 1 2. g(x) = (x − 4)^2 3. g(x) = (x + 2)^2 + 3 4. g(x) = (x − 3)^2 − 4

1. Up 1 2. Right 1 3. Left 2, Up 3 4. Right 3, Down 4

How many solutions (x-intercepts) are there when the discriminant is greater than 0?

2 real solutions (x-intercepts)

A ball is hit straight up into the air. The height of the ball in meters is given by the function h(t)= -5(t-3)^2+45 t seconds after the ball is hit. In how many seconds will the ball hit the ground?

6 seconds

Use transformations of the parent quadratic function to determine the vertex and axis of symmetry of the graph of each function. 7. g(x) = (x − 8)^2 8. g(x) = (x + 6)^2 − 4

7. Vertex= (8,0) AOS= x=8 8. Vertex= (-6,-4) AOS= x=-6

Write a quadratic function g(x) that represents each transformation of the function f(x) = x^2 9. translate 6 units right 10. translate 10 units down 11. translate 9 units right and 6 units up 12. translate 4 units left and 8 units down

9. (x-6)^2 10. x^2-10 11. (x-9)^2 +6 12. (x+4)^2-8

When will the graph shrink?

When a is less than 1 but greater than 0

When will the parabola reflect over the x-axis?

When a is negative (-)

For each function, identify all transformations of the function f(x) = x2 Then graph the function. 14. g(x)= -1/3x^2 15. g(x)= 1/5x^2 16. g(x)= 1/2(x-3)^2 17. g(x) = −2(x + 3)2 + 1 18. g(x) = −3(x + 2)2 − 5

You got this!

Graph this parabola: y=(x+4)^2

You got this!

The function h(t) = −16t^2 + 22t + 4 models the height h in feet of a football t seconds after it is thrown. a. Write the function in vertex form. b. To the nearest foot, what is the greatest height that the football reaches? Explain your answer. c. To the nearest tenth of a second, how long after the football is thrown does it reach its greatest height? Explain your answer.

You got this!

Write each function in vertex form. Then describe the transformation(s) from the parent function and use the transformations to graph the function. 26. g(x) = x2 − 4x − 1 27. g(x) = −2x2 + 12x − 17 28. g(x) = 3x2 + 6x + 1

You got this!

Write each function in vertex form. Then identify the vertex and axis of symmetry of the function's graph, and tell which direction the graph opens. 29. f(x) = x2 − 16x + 71 30. f(x) = 2x2 + 36x + 142 31. f(x) = −3x2 + 6x + 9 32. f(x) = x2 − 2x + 5

You got this!

Factored form

a(x-p)(x-q); where p and q are the x-intercepts

What are the values for the parameters (a,h,k) for the function f(x)=-2(x+3)^2 +1

a=-2, h=-3, k=1

Vertical Translation

f(x)+k (+ up, - down)

Write a function that includes the translation of 3 units to the left

f(x)= (x+3)^2 -1

From standard form, how would you derive the vertex?

x= -b/2a; then plug x into function to find y

The parabola y=x^2 is shifted down by 3 units and to the left by 2 units. What is the equation of the new parabola?

y= (x+2)^2 -3

The parabola y=x^2 is shifted down by 6 units and to the right by 5 units. What is the equation of the new parabola?

y= (x-5)^2-6

Consider a parabola P that is congruent to y=x^2, opens upward, and has a vertex at (-1,3). Now find the equation of a new parabola that results of P if it is reflected over the x-axis and translated 3 units down.

y= -(x+1)^2

What is the equation of a parabola that goes through point (2,-1)?

y= -1/4x^2

What is the equation of a parabola that goes through point (3,-6)?

y= -2/3x^2

The parabola y=x^2 is reflected across the x-axis and then scaled vertically by a factor of 5. What is the equation of the new parabola?

y= -5x^2

What is the equation of a parabola that goes through point (4,8)?

y= 1/2x^2

Write an equation that shows a translation of 3 units to the left and a vertical compression by a factor of 2 to the graph of y=x^2.

y= 2(x+3)^2

Create three different parabolas that pass through the points (-4,0) and (3,0).

y=(x+4)(x-3) y=-2(x+4)(x-3) y=1/4(x+4)(x-3)

Write an equation that opens downward, is equal to y=x^2, and that has a vertex at (0,3).

y=-x^2+3

What parabola equation goes through the points (6,0), (0,6), and (4,-2)?

y=1/2(x-4)^2-2

From standard form, how would you graph? (3 ways)

Change it into vertex form or Find the x of the vertex (x=-b/2A) and solve for y. Then, in standard form c is the y-intercept or Create a table of values

Which of these functions has the widest graph when they are graphed on the same coordinate plane? A. f (x) = −2x^2 B. f (x) = 5x^2 C. f(x)= 1/2x^2 D. f(x)= -1/5x^2

D. f(x)= -1/5x^2

f(x)= x^2 g(x)​=(x+4)^2−1​ We can think of g(x) as a translated (shifted) version of f(x). Describe the transformation

Down by 1, left by 4

Describe the transformations then sketch the graph of each function: f(x)= -(x-4)^2-1

Down by 1, to the right by 4, reflected over x-axis, no stretch/shrink

From factored form, how would you find the AOS?

Find the AOS by averaging the x-intercepts

From factored form, how would you find the x-intercepts?

Find the x-intercepts by solving for 0

For the rest of practice, retake the Unit 2B Review

Good Luck!

From standard form, how would you find the y-intercept?

It is the constant c.

DEFINITIONS

NEXT CARDS

How many solutions (x-intercepts) are there when the discriminant is less than 0?

No real solutions (x-intercepts)

How many solutions (x-intercepts) are there when the discriminant is equal to 0?

One real solution (x-intercepts)

Quadratic Pattern

Over (x) Up (y) 1 1 2 4 3 9

From factored form, how would you find the vertex?

Plug the AOS into the original function to find the vertex

From standard form, how would you find the x-intercepts?

Solve for 0

f(x)= x^2 g(x)​=x^2−8​ We can think of g of x as a translated (shifted) version of f of x. Complete the description of the transformation. "To get the function g(x), shift f(x)_______by______"

To get the function g(x), shift f(x) DOWN by 8

f(x)= x^2 g(x)​=(x+3)^2+5​ We can think of g(x) as a translated (shifted) version of f(x). Describe the transformation.

Up by 5, Left by 3

When will the graph stretch?

When a is greater than 1

Vertex Form

f(x)= a(x-h)^2 +k (h,k) = vertex

Horizontal Translation

f(x+h) (+ left, - right)

The function g(x) is a translation of f(x) = x^2 The vertex of the graph of g(x) is (−4, 7). What is the equation of g(x)? Explain your answer.

g(x)= (x+4)^2+7

List the sequences of steps that are required to graph the function f(x)=-(x+4)^2-6

horizontal translation of 4 to the left, then a reflection over the x-axis, and finally, a vertical translation of 6 down.

What is the h in the vertex equation?

it is a translation to the right (-) or the left (+)

What is the k in the vertex equation?

it is a translation up (+) or down (-)

Parabola

the graph of a quadratic function (u-shaped)

Vertex

the point in the middle of the parabola (min/max) that lies on the axis of symmetry

What is the a in the vertex equation?

the sign will tell us if we need to reflect the value (-) or stretch/shrink it

Axis of symmetry

the vertical line through the vertex

From standard form, how would you find the AOS?

x= -b/2a

The parabola y=x^2 is scaled vertically by a factor of 2/3 What is the equation of the new parabola?

y= 2/3x^2

y= -5(x-2)^2+20 What are the coordinates of the vertex?

(2, 20)