Solve \((x-1)^2 = 16\). Explain or show your reasoning.

Here is one way to solve the equation \(\frac59 y^2 = 5\). Explain what is done in each step.

\(\begin{align}\frac59y^2 &= 5 &\quad &\text{Original equation}\\5y^2&=45 &\quad &\text{Step 1} \\\\y^2&=9 &\quad& \text{Step 2} \\\\y=3 \qquad &\text{or} \qquad y=\text-3 &\quad& \text{Step 3} \end{align}\)

Diego and Jada are working together to solve the quadratic equation \((x-2)^2 = 100\).

Jada asks Diego why he divides each side by 2 and he says, “I want to find a number that equals 100 when multiplied by itself. That number is half of 100.”

1. What mistake is Diego making?
2. If you were Jada, what could you say to Diego to help him realize his mistake?

A billboard installer accidentally drops a tool while working on a billboard. The height of the tool \(t\) seconds after it is dropped is given by the function \(h(t) = 115-16t^2\), where \(h\) is in feet.

1. Find \(h(2)\). Explain what this value means in this situation.
2. Find \(h(0)\). What does this value tell us about the situation?
3. Is the tool still in the air 4 seconds after it is dropped? Explain how you know.

A zoo offers unlimited drink refills to visitors who purchase its souvenir cup. The cup and the first fill cost \$10, and refills after that are \$2 each. The expression \(10+2r\) represents the total cost of the cup and \(r\) refills.

1. A family visited the zoo several times over a summer. That summer, they paid \$30 for one cup and multiple refills. How many refills did they buy?
2. A visitor has \$18 to spend on drinks at the zoo today and buys a souvenir cup. How many refills can they afford during the visit?
3. Another visitor spent \$10 on this deal. Did they buy any refills? Explain how you know.

Clare is 5 years older than her sister.

1. Write an equation that defines her sister's age, \(s\), as a function of Clare’s age, \(c\).
2. Write an equation that defines Clare’s age, \(c\), as a function of her sister's age, \(s\).
3. Graph each function. Be sure to label the axes.

   

   

4. Describe how the two graphs compare.

The graph shows a model for the weight of snow as it melts. The weight decreases exponentially.

1. By what factor does the weight of the snow decrease each hour? Explain how you know.

   

   Graph of a function on grid, origin O. Horizontal axis, time, hours, from 0 to 6 by 0 point 5's. Vertical axis, weight, kilograms, from 0 to 18 by 2's. Plotted coordinates as follows: 0 comma 15, 1 comma 6, 2 comma 2 point 4, 3 comma 0 point 96, 4 comma 0 point 384, 5 comma 0 point 1536, 6 comma 0 point 06144.  

2. Does the model predict that the weight of the snow will reach 0 kg? Explain your reasoning.
3. Will the weight of the actual snow, represented by the graph, reach 0? Explain how you know.