This question provides an example of an initial bank account investment with a fixed return and tests the student's understanding of an application of intercepts.

", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["p", "friend", "m", "c", "bal"], "type": "question", "extensions": ["geogebra", "random_person"], "advice": "The balance increases by £{m} each year, so $m = \\var{m}$.

\nWe know that at $x = 9$ years, $y = \\var{bal}$. Substituting these values into the equation, we obtain

\n\\[ \\var{bal} = \\var{m} \\times 9 + c \\]

\nRearrange this to find $c$:

\n\\begin{align}

c &= \\var{bal} - \\var{m} \\times 9 \\\\

&= \\var{bal} - \\var{m*9} \\\\

&= \\var{c}

\\end{align}

So the formula for the account balance is

\n\\[y = \\var{m}x+\\var{c}\\text{.}\\]

\nThe constant term $\\var{c}$ determines the point at which the line crosses the $y$-axis. This point is called the *$y$-intercept*.

The initial investment is the value of $y$ at $x = 0$, so it's £{c}$.

\nIt is useful to plot the graph of {friend['name']}'s savings account against your own for comparison ({friend['name']}'s balance is shown as a dashed line):

\n{geogebra_applet('gpFmg3Ex',[[\"p\",p],[\"m\",m],[\"c\",c]])}

\nUsing this graph, we can see that only two of the statements are true:

\n- \n
- The two lines are parallel. This is because they both increase by £{m} each year. \n
- The plot of {friend['name']}'s balance has a higher $y$-intercept because {friend['pronouns']['their']} initial balance is higher. \n

As the gradients of the two lines on the graph are the same, we can eliminate the other two statements about the lines converging and about having a higher gradient.

", "variable_groups": [], "rulesets": {}, "statement": "You are a forgetful investor set on saving enough money to buy a new fishing boat for when you retire.

\nYour savings account manager tells you your savings account is worth £{formatnumber(bal,\"en\")}.

\nYou have forgotten the principal amount you started with the account with; however you do know that you have been saving for exactly nine years now and your manager informs you that the bank has been paying you a premium of £{m} per year.

\nYour account manager shows you this graph, which plots account balance over time for a given principal amount.

\n*The line on the graph below can be repositioned by dragging the slider.*

{geogebra_applet('HtnCWSSQ',[[\"p\",p],[\"m\",m],[\"c\",c]])}

", "name": "Applied y-intercepts: Investing in boats", "parts": [{"scripts": {}, "gaps": [{"showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{m}x+{c}", "scripts": {}, "variableReplacements": [], "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}], "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "With $y$ representing the account balance and $x$ the number of years the account has been open, give an expression for the balance in the form $y=mx+c$.

\n$y=$ [[0]]

\n", "variableReplacementStrategy": "originalfirst"}, {"shuffleChoices": true, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "prompt": "

Which of the following elements of the graph corresponds to the constant part of the equation?

", "showFeedbackIcon": true, "displayColumns": 0, "choices": ["y-intercept

", "x-intercept

", "z-intercept

", "The origin

"], "scripts": {}, "distractors": ["", "", "", ""], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "marks": 0, "variableReplacements": [], "matrix": ["1", 0, 0, 0], "displayType": "dropdownlist"}, {"scripts": {}, "gaps": [{"showpreview": true, "vsetrangepoints": 5, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "showFeedbackIcon": true, "checkvariablenames": false, "type": "jme", "answer": "{c}", "scripts": {}, "variableReplacements": [], "checkingtype": "absdiff", "vsetrange": [0, 1], "marks": 1, "expectedvariablenames": []}], "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "From your answer to the last question, state the initial investment you made towards saving for your new fishing boat.

\nInitial investment $=$ $£$[[0]].

", "variableReplacementStrategy": "originalfirst"}, {"shuffleChoices": true, "type": "m_n_2", "maxAnswers": 0, "minMarks": 0, "showCorrectAnswer": true, "minAnswers": 0, "prompt": "Your friend, {friend['name']}, is considerably wealthier than you are, so {friend['pronouns']['they']} {if(friend['gender']='neutral','start','starts')} with twice the investment you did but still {if(friend['gender']='neutral','receive','receives')} the same annual payment of $£\\var{m}$.

\nWhich of the following statements comparing the graph of {friend['name']}'s account balance to yours are true?

", "showFeedbackIcon": true, "displayColumns": "1", "choices": ["The gradient is equal and hence {friend['pronouns']['their']} line would be parallel to yours.

", "The plot of {friend['pronouns']['their']} balance crosses the $y$-axis at a higher point than yours.

", "The plot of {friend['pronouns']['their']} balance has a higher gradient.

", "The plots of your balance and {friend['pronouns']['theirs']} cross at some point.

"], "scripts": {}, "distractors": ["", "", "", ""], "maxMarks": 0, "variableReplacementStrategy": "originalfirst", "marks": 0, "variableReplacements": [], "matrix": ["1", "1", 0, 0], "displayType": "checkbox", "warningType": "none"}], "tags": ["applications of y-intercepts", "taxonomy", "y-intercept"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"bal": {"description": "", "group": "Ungrouped variables", "definition": "m*9+c", "name": "bal", "templateType": "anything"}, "c": {"description": "", "group": "Ungrouped variables", "definition": "random(995,1005,1010,1015,1020,1025)", "name": "c", "templateType": "anything"}, "p": {"description": "not intercept, starting intercept

", "group": "Ungrouped variables", "definition": "1100", "name": "p", "templateType": "anything"}, "friend": {"description": "", "group": "Ungrouped variables", "definition": "random_person()", "name": "friend", "templateType": "anything"}, "m": {"description": "", "group": "Ungrouped variables", "definition": "random(15,30,45)", "name": "m", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}]}