# Lesson 9

Standard Form and Factored Form

- Let’s write quadratic expressions in different forms.

### Problem 1

Write each quadratic expression in standard form. Draw a diagram if needed.

- \((x+4)(x-1)\)
- \((2x-1)(3x-1)\)

### Problem 2

Consider the expression \(8 - 6x + x^2\).

- Is the expression in standard form? Explain how you know.
- Is the expression equivalent to \((x-4)(x-2)\)? Explain how you know.

### Problem 3

Which quadratic expression is written in standard form?

\((x+3)x\)

\((x+4)^2\)

\(\text-x^2-5x+7\)

\(x^2+2(x+3)\)

### Problem 4

Explain why \(3x^2\) can be said to be in both standard form and factored form.

### Problem 5

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?

\(y=16t^2\)

\(y=48t\)

\(y=180-16t^2\)

\(y=180-48t\)

### Problem 6

A football player throws a football. The function \(h\) given by \(h(t)=6+75t-16t^2\) describes the football’s height in feet \(t\) seconds after it is thrown.

Select **all **the statements that are true about this situation.

The football is thrown from ground level.

The football is thrown from 6 feet off the ground.

In the function, \(\text-16t^2\) represents the effect of gravity.

The outputs of \(h\) decrease then increase in value.

The function \(h\) has 2 zeros that make sense in this situation.

The vertex of the graph of \(h\) gives the maximum height of the football.

### Problem 7

*Technology required*. Two rocks are launched straight up in the air.

- The height of Rock A is given by the function \(f\), where \(f(t)=4+30t-16t^2\).
- The height of Rock B is given by function \(g\), where \(g(t)=5+20t-16t^2\).

In both functions, \(t\) is time measured in seconds and height is measured in feet. Use graphing technology to graph both equations.

- What is the maximum height of each rock?
- Which rock reaches its maximum height first? Explain how you know.

### Problem 8

The graph shows the number of grams of a radioactive substance in a sample at different times after the sample was first analyzed.

- What is the average rate of change for the substance during the 10 year period?
- Is the average rate of change a good measure for the change in the radioactive substance during these 10 years? Explain how you know.

### Problem 9

Each day after an outbreak of a new strain of the flu virus, a public health scientist receives a report of the number of new cases of the flu reported by area hospitals.

time since outbreak in days | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|

number of new cases of the flu | 20 | 28 | 38 | 54 | 75 | 105 |

Would a linear or exponential model be more appropriate for this data? Explain how you know.

### Problem 10

\(A(t)\) is a model for the temperature in Aspen, Colorado, \(t\) months after the start of the year. \(M(t)\) is a model for the temperature in Minneapolis, Minnesota, \(t\) months after the start of the year. Temperature is measured in degrees Fahrenheit.

- What does \(A(8)\) mean in this situation? Estimate \(A(8)\).
- Which city has a higher predicted temperature in February?
- Are the 2 cities’ predicted temperatures ever the same? If so, when?