{"id":"cb_question_bank:6f817f7a-a5b8-43e7-91e5-f6232f3e2886","source_id":"cb_question_bank","source_item_id":"6f817f7a-a5b8-43e7-91e5-f6232f3e2886","external_id":"baa55cdd-6e64-416a-8f18-d484e43741e8","ibn":null,"question_id":"1f353a9e","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":6,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"f left parenthesis t right parenthesis equals 8,000 left parenthesis 0.65 right parenthesis Superscript t The given function f models the number of coupons a company sent to their customers at the end of each year, where t represents the number of years since the end of 1998 , and 0 less than or equals t less than or equals 5 . If y equals f left parenthesis t right parenthesis is graphed in the ty -plane, which of the following is the best interpretation of the y -intercept of the graph in this context?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: center;\"><math alttext=\"f left parenthesis t right parenthesis equals 8,000 left parenthesis 0.65 right parenthesis Superscript t\"><mi>f</mi><mfenced><mi>t</mi></mfenced><mo>=</mo><mn>8,000</mn><msup><mfenced><mn>0.65</mn></mfenced><mi>t</mi></msup></math></p>\n<p style=\"text-align: left;\">The given function <math alttext=\"f\"><mi>f</mi>\n</math> models the number of coupons a company sent to their customers at the end of each year, where <math alttext=\"t\"><mi>t</mi>\n</math> represents the number of years since the end of <math alttext=\"1998\"><mn>1998</mn></math>, and <math alttext=\"0 less than or equals t less than or equals 5\"><mn>0</mn><mo>≤</mo><mi>t</mi><mo>≤</mo><mn>5</mn></math>. If <math alttext=\"y equals f left parenthesis t right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>t</mi></mfenced></math> is graphed in the <em>ty</em>-plane, which of the following is the best interpretation of the <em>y</em>-intercept of the graph in this context?</p>","rationale_html":"<p style=\"text-align: left;\">Choice D is correct. The <em>y</em>-intercept of a graph in the <em>ty</em>-plane is the point where <math alttext=\"t equals 0\"><mrow>\n\t<mi>t</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math>. For the given function <math alttext=\"f\"><mi>f</mi></math>, the<em>&nbsp;y</em>-intercept of the graph of <math alttext=\"y equals f left parenthesis t right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>t</mi></mfenced></math> in the <em>ty</em>-plane can be found by substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"t\"><mi>t</mi>\n</math> in the equation <math alttext=\"y equals 8,000 left parenthesis 0 period 65 right parenthesis Superscript t\"><mi>y</mi><mo>=</mo><mn>8,000</mn><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>65</mn></mrow></mfenced><mi>t</mi></msup></math>, which gives <math alttext=\"y equals 8,000 left parenthesis 0 period 65 right parenthesis Superscript 0\"><mi>y</mi><mo>=</mo><mn>8,000</mn><msup><mfenced><mrow><mn>0</mn><mo>.</mo><mn>65</mn></mrow></mfenced><mn>0</mn></msup></math>. This is equivalent to <math alttext=\"y equals 8,000 left parenthesis 1 right parenthesis\"><mi>y</mi><mo>=</mo><mn>8,000</mn><mfenced><mn>1</mn></mfenced></math>, or <math alttext=\"y equals 8,000\"><mrow>\n\t<mi>y</mi>\n\t<mo>=</mo>\n\t<mn>8,000</mn>\n</mrow>\n</math>. Therefore, the <em>y</em>-intercept of the graph of <math alttext=\"y equals f left parenthesis t right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>t</mi></mfenced></math> is <math alttext=\"left parenthesis 0 comma 8,000 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>8,000</mn></mrow></mfenced></math>. It’s given that the function&nbsp;<math alttext=\"f\"><mi>f</mi></math> models the number of coupons a company sent to their customers at the end of each year. Therefore,&nbsp;<math alttext=\"f left parenthesis t right parenthesis\"><mi>f</mi><mfenced><mi>t</mi></mfenced></math> represents the estimated number of coupons the company sent to their customers at the end of each year. It's also given that&nbsp;<math alttext=\"t\"><mi>t</mi>\n</math> represents the number of years since the end of <math alttext=\"1998\"><mn>1998</mn></math>. Therefore, <math alttext=\"t equals 0\"><mrow>\n\t<mi>t</mi>\n\t<mo>=</mo>\n\t<mn>0</mn>\n</mrow>\n</math> represents <math alttext=\"0\"><mn>0</mn>\n</math> years since the end of <math alttext=\"1998\"><mn>1998</mn></math>, or the end of <math alttext=\"1998\"><mn>1998</mn></math>. Thus, the best interpretation of the <em>y</em>-intercept of the graph of&nbsp;<math alttext=\"y equals f left parenthesis t right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>t</mi></mfenced></math> is&nbsp;that the estimated number of coupons the company sent to their customers at the end of <math alttext=\"1998\"><mn>1998</mn></math> was <math alttext=\"8,000\"><mn>8,000</mn>\n</math>.</p>\n<p>Choice A is incorrect and may result from conceptual or calculation errors.&nbsp;</p>\n<p>Choice B is incorrect and may result from conceptual or calculation errors.&nbsp;</p>\n<p>Choice C is incorrect and may result from conceptual or calculation errors.&nbsp;</p>","correct_answer":["D"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.826000+00:00","source_updated_at":"2023-08-02T20:25:59.826000+00:00","options":[{"label":"A","content_html":"<p style=\"text-align: left;\">The minimum estimated number of coupons the company sent to their customers during the <math alttext=\"5\"><mn>5</mn>\n</math> years was <math alttext=\"1,428\"><mn>1,428</mn>\n</math>.</p>","ord":0},{"label":"B","content_html":"<p style=\"text-align: left;\">The minimum estimated number of coupons the company sent to their customers during the <math alttext=\"5\"><mn>5</mn>\n</math> years was <math alttext=\"8,000\"><mn>8,000</mn>\n</math>.</p>","ord":1},{"label":"C","content_html":"<p style=\"text-align: left;\">The estimated number of coupons the company sent to their customers at the end of&nbsp;<math alttext=\"1998\"><mn>1998</mn></math> was <math alttext=\"1,428\"><mn>1,428</mn>\n</math>.</p>","ord":2},{"label":"D","content_html":"<p style=\"text-align: left;\">The estimated number of coupons the company sent to their customers at the end of&nbsp;<math alttext=\"1998\"><mn>1998</mn></math> was <math alttext=\"8,000\"><mn>8,000</mn>\n</math>.</p>","ord":3}],"attribution":{"source_id":"cb_question_bank","source_name":"College Board digital SAT question bank (community dump)","rights_holder":"College Board","canonical_url":"https://satsuitequestionbank.collegeboard.org/","retrieved_from":"https://raw.githubusercontent.com/mdn522/sat-question-bank/main/data/cb-digital-questions.json","attribution":"SAT practice questions © College Board, from the official SAT Suite Question Bank. Retrieved via the community dump at github.com/mdn522/sat-question-bank.","disclaimer":"SAT® and PSAT/NMSQT® are trademarks registered by the College Board, which is not affiliated with, does not sponsor, and does not endorse this project.","license":"College Board copyright; source repository carries no licence. Local use only.","redistributable":false,"item_count":2017}}