Quadratics

what is the y-intercept of the graph of the equation y=x2-5x+4? 2.4

(0,4) (4,0) (1,0) switch signs

A company sells a video game. If the price of the game in dollars is p the company estimates that it will sell 20,000-500p games. Write a function that represents the revenue, r, in dollars from selling games if the game is priced at p dollars? 2.4

(R(p) (PI=p(2000-500p)

Technology required Two rocks are launched straight up in the air. The height of rock A is given by the function f, were f (t)= 4+30t-16t2. The height of rock B is given by g, were g(t)=5+20t-16t2. Using graphing technology to graph both equations. Determine which rock hits the ground first explain how you know 2.2

(g(t) hits the ground first. It starts 1 foot higher but has a 10 ft/sec slower speed. The graph crosses the x-axis after 1.46 sec.

Quadratic part 2 summary 2.3

A quadratic function can often be defined by many different but equivalent expressions. For example, we saw earlier that the predicted revenue, in thousands of dollars, from selling a downloadable movie at x dollars can be expressed with x(18-x), which can also be written as 18x-x2. The former is a product of x and 18-x, and the latter is a difference of 18x and x2, but both expressions represent the same function.

The graph of y=x2 is a parabola opening upward with vertex at (0,0). Adding a constant term 5 gives y=x2+5 and raises the graph by 5 units. Subtracting 4 from x2 gives y=x2-4 and moves the graph 4 units down. 2.5

A table of values can help us see that adding 5 to x2 increases all the output values of y=x2 by 5, which explains why the graph moves up 5 units. Subtracting 4 from x2 decreases all the output values of y=x2 by 4, which explains why the graph shifts down by 4 units.

A(t) is the average high temperature in Aspen, Colorado, t months after the start of the year. M(t) is the tempter in Minneapolis, Minnesota, t months after the start of the year a. What does A(8) mean in this situation? Estimate A(8) b. Which city had a higher average template in February? 2.4

A(8) is the highest's tempuater in months in Aspen Colorado b. Aspen

1. A rocket is launched in the air and it's height, in feet, is modeled by the function h. Here is the graph representing h. Select all the true statements about the situation 2.2

A) the rocket is launched from a height less than 20 feet above the ground C) The rocket reaches its maximum height after about 3 seconds e) The maximum height if the rocket is about 160 feet

Slandered form 2,5

Ax^2+bx+c

3. What do we need to sketch a graph? a. The functions f,g, and h are given. Predict the x-intercepts and the x-coordinate of the vertex of each function 2.5

Equation f(x)=(x+3)(x-5)=0 x-intercepts (-3,0) (5,0) x-coordinate of the vertex-3+5/2 =-5 equation g(x)=2x(x-3) x-intercepts (0,0) (3,0) equation h(x)=(x+4)(4-x)=0 4-4 (-4,0) (4,0) x coordinate of the vertex -4+4/2=-2

2.5 Graphing in Factored and Standard Form Summaries Factored Form The function f given by f(x)=(x+1)(x-3) is written in factored form. Recall that this form is helpful for finding the zeros of the function (where the function has the value 0) and telling us the x-intercepts on the graph representing the function. 2.5

Here is a graph representing f. It shows 2 x-intercepts at x=-1 and x=3. If we use -1 and 3 as inputs to f, what are the outputs? f(-1)=(-1+1)(-1-3)=(0)(-4)=0 f(3)=(3+1)(3-3)=(4)(0)=0

Multiplying x2 by a number greater than 1 makes the graph steeper, so the parabola is narrower than that representing x2. Multiplying x2 by a number less than 1 but greater than 0 makes the graph less steep, so the parabola is wider than that representing x2. Multiplying x2 by a number less than 0 makes the parabola open downward. 2.5

If we compare the output values of 2x2 and -2x2, we see that they are opposites, which suggests that one graph would be a reflection of the other across the x-axis.

Quadratic summary 2, 2.3

It makes sense that the revenue goes down after a certain point, since if the price is too high nobody will buy a ticket. From the graph, we can tell that the greatest revenue, $1,200, comes from selling the tickets for $4 each. We can also see that the domain of the function r is between 0 and 8. This makes sense because the cost of the tickets can't be negative, and if the price were more than $8, the model does not work, as the revenue collected cannot be negative. (A negative revenue would mean the number of tickets sold is negative, which is not possible.)

1. Here is a graph of a function w defined by w(x)=(x+1.6) (x-2). Three points on the graph a labeled. Find the values. 2.5

Multiply the constant y-intercept a,b=(1.60, 0) c,d=(2,0) e,f=(-3.2) x=0 w(0)=(0+-1.6) (0,-2) w(0)=(1.6) (-2) w(0)=-3.2

quadrilateric part 2 summary 3, 2.3

Multiplying (x+2) and (x+3) gives the area of the rectangle. Adding the areas of the four sub-rectangles also gives the area of the rectangle. This means that (x+2)(x+3) is equivalent to x2+2x+3x+6, or to x2+5x+6.Notice that the diagram illustrates the distributive property being applied. Each term of one factor (say, the x and the 2 in x+2) is multiplied by every term in the other factor (the x and the 3 in x+3).

Lesson Summary 2.2

Notice that the distance fallen is increasing each second. The average rate of change is increasing each second, which means the cannonball is speeding up over time. This comes from the influence of gravity, which is influenced by the quadratic expression 16t2. It is the exponent 2 in that expression that makes it increase by larger and larger amounts. We also looked at the height of objects that are launched upward and then come back down because of gravity. The expression -16t2 represents the effect of gravity, which eventually causes the object to slow down, stop and start falling back again.

Standard Form 2.5

Remember that the graph representing any quadratic function is a shape called a parabola. People often say that a parabola "opens upward" when the lowest point on the graph is the vertex (where the graph changes direction), and "opens downward" when the highest point on the graph is the vertex. Each coefficient in a quadratic expression written in standard form ax2+bx+c tells us something important about the graph that represents it.

Quadratic part 2 summary 2, 2.3

Sometimes a quadratic expression is a product of two factors that are each a linear expression, for example (x+2)(x+3). We can write an equivalent expression by thinking about each factor, the (x+2) and (x+3), as the side lengths of a rectangle, and each side length decomposed into a variable expression and a number.

Quadractic summary 2.3

Suppose we are selling raffle tickets and deciding how much to charge for each ticket. When the price of the tickets is higher, typically fewer tickets will be sold.Let's say that with a price of d dollars, it is possible to sell 600-75d tickets. We can find the revenue by multiplying the price by the number of tickets expected to be sold. A function that models the revenue r collected is r(d)=d(600-75d). Here is a graph that represents the function.

In general, the constant term of a quadratic expression in standard form influences the vertical position of the graph. An expression with no constant term (such as x2 or x2+9x) means that the constant term is 0, so the y-intercept of the graph is on the x-axis. It's not shifted up or down relative to the x-axis. 2.5

The coefficient of the squared term in a quadratic function also tells us something about its graph. The coefficient of the squared term in y=x2 is 1. Its graph is a parabola that opens upward.

Lesson Summary 2.2

The distance traveled by a falling object in a given amount of time is an example of an quadratic function. Galileo is said to have dropped balls of different mass from leaning over the tower of Pisa, which is about 190 feet tall, to show that they traveled the same distance in the same time. In fact the equation d=16t2 models the distance d, in feet, that the cannonball falls after t seconds, no matter what its mass. Because 16x42=256, and the tower is only 190 feet tall, the ball hits the ground before 4 seconds

Because the inputs -1 and 3 produce an output of 0, they are the zeros of the function f. And because both x values have 0 for their y value, they also give us the x-intercepts of the graph (the points where the graph crosses the x-axis, which always have a y-coordinate of 0). So, the zeros of a function have the same values as the x-coordinates of the x-intercepts of the graph of the function. 2.5

The factored form can also help us identify the vertex of the graph, which is the point where the function reaches its minimum value. Notice that the x-coordinate of the vertex is 1, and that 1 is halfway between -1 and 3. Once we know the x-coordinate of the vertex, we can find the y-coordinate by evaluating the function: f(1)=(1+1)(1-3)=2(-2)=-4. So the vertex is at (1,-4).

Parabola 2.5

The shape of the graph of a quadratic function

Each expression represents an objects distance from the ground in meters as a function of time, t, in seconds Object A; -5t2+25t+50 object B -5t2 + 50t +25 2.2

Which item was launched at the greatest vertical speed? Object B 50t vs 25t Which object was launched from the greats height Object A 50 vs 25

A football player throws a football. The function h given by h(t)=6+75t-16t2 describes the football's height in feet t seconds after it is thrown select all statements that are true about this situation 2.4

a the football is thrown from the ground level b. the football is thrown from 6 feet off the grown c. the function -16t2 represents the effect of gravity d. the outputs of h decrease then increase in value e. the function h has 2 zeros that make sense in this situation f. the vertex of the graph of h gives the maximum height of the football b and c 6 vertex starting 0 not on graph

A small ball is dropped from a tall building Which equation could represent the ball's height, h, in feet, relative to the ground, as a function of time, t, in seconds? 2.2

a) h=100-16t not squared b) h=100-16t2 Quadratic 16t2=gravity

Determine whether 5n2 or 3n will have the greater value when 2.2

a) n=1 5, 3 5n2 b)n=3 45,25 5n2 c)n=5 125, 243 3n Larger value

7. What are the x intercepts of the graph of the function defined by (x-2) (2x+1)? 2.4

a. (2,0) and (-1,0) b.(2,0) and (-1/2,0) c. (-2,0) and (1,0) d.(-2,0) and (1/2,0) d is the correct answer Divide by 2 2(x+1/2) or 2x+1=0 -1 -1 2x/2=-1/2 x=1/2

Jada dropped her sunglasses from a bridge over a river. Which equation could represent the distance y fallen in feet as a function of time, t, in seconds? 2.4

a.y=16t2 b. y=48 c. y= 180-16t2 d. y=180-48t C is correct because 16t2 d(t)=

a.b=0 2.5

a=0 b=0 b=anything a=anything

b. Plot the points from the tables on the same coordinate plane. (Don't worry about points off the grid.) x-5(-5+4) y-5(-5-4) calculator. C. Make a couple of observations about how the two graphs compare. d. Where would the x-intercept and vertex be for the equation h(x)=-x(x-10) 2.5

b. x -5 -4 -3 -2 1 0 1 2 3 4 5 f(x) 5 0 -3 -4 -3 0 5 12 21 32 45 g(x)= 45 32 21 12 5 0 -3 -4 -3 0 5 c. notice that they were the same graph just shifted over d. Vertex is always going to be in-between x-intercept x int. (10,0) y (0,0) vertex (5,25) 0+10/2=5 -5(5-10)=25

Here is a graph that represents a quadratic function. Which expression could define this function? a.(x+3) (x+1) b.(x+3) (x-1) c. (x-3) (x+1) d.(x-3) (x-1) change 2.4

c is correct (x-3) (x+1)

Select all of the expressions that give the number of small squares in step n 2.2

c) n(n+1) d) n2+n step 1 n, n, 1, n step 2 2,2,1,2 step 3 3,3,1,3 n, n+1, n2 n

When a quadratic function is in standard form, the y-intercept is clear: its y-coordinate is the constant term c 2.5

in ax2+bx+c. To find the y-intercept from factored form, we can evaluate the function at x=0, because the y-intercept is the point where the graph has an input value of 0. f(0)=(0+1)(0-3)=(1)(-3)=-3.

A basketball travels d meters t seconds after being dropped from the top of a building. The distance traveled by the baseball can be molded by equation d=5t2 2.2

t (seconds) 0, 0.5, 1, 1.5, 2 d (meters) 0, 1.25, 5, 11.25, 20 Is the baseball traveling at a constant speed? No, its not quadratic. No common difference

2. Comparing 2 graphs Two functions are f(x)=x(x+4) and g(x)=x(x-4) a. complete the table of values for each function. Then, determine the x-intercepts and vertex of each graph. Be prepared to explain how you know. Ones positive ones negative whenever thesers an x=0 2.5

x intercepts; f(x) (-4,0) (0,0) g(x) (4,0) (0,0)

2.4 Consider the expression 8-6x+x2 a. Is the expression in slandered form? b. Is the expression equivalent to (x-4)(x-2)? Explain how you know

x2-2x-4x+8 (x-2)2 = (x-2(x-2) multiply connect

Quadratic 2.5

x^2

A quadratic function f is defined by f(x)=(x-7) (x+3) a. without graphing identify the x intercept and y intercept of the graph of f. Explain how to you know 2.4

y(0,7) x (-3,0) y-intercept (0,-21) (7-7) (7+3)=0 0x10 (0,-7) (0+3) -7+3=-21