{"id":"cb_question_bank:b57d05b0-6612-4c46-b6db-4981640c626b","source_id":"cb_question_bank","source_item_id":"b57d05b0-6612-4c46-b6db-4981640c626b","external_id":"1456688b-dc2f-4218-865c-e9bd32de6518","ibn":null,"question_id":"de39858a","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":7,"answer_type":"mcq","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The function h is defined by h ( x ) = a x + b , where a and b are positive constants. The graph of y = h ( x ) in the x y -plane passes through the points ( 0 , 10 ) and ( -2 , 325 36 ). What is the value of a b ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The function <math><mi>h</mi>\n</math> is defined by <math><mrow><mi>h</mi><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mrow><mi>x</mi></mrow><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup><mo>+</mo><mi>b</mi></mrow></math>, where <math><mi>a</mi>\n</math> and <math><mi>b</mi>\n</math> are positive constants. The graph of <math><mrow><mi>y</mi><mo>=</mo><mi>h</mi><mrow><mo stretchy=\"true\" fence=\"true\" form=\"prefix\">(</mo><mrow><mi>x</mi></mrow><mo stretchy=\"true\" fence=\"true\" form=\"postfix\">)</mo></mrow></mrow></math> in the <math><mrow>\n\t<mi>x</mi>\n\t<mi>y</mi>\n</mrow>\n</math>-plane passes through the points (<math><mn>0</mn>\n</math>, <math><mn>10</mn>\n</math>) and (<math><mn>-2</mn>\n</math>, <math><mfrac>\n\t<mn>325</mn>\n\t<mn>36</mn>\n</mfrac>\n</math>). What is the value of <math><mrow>\n\t<mi>a</mi>\n\t<mi>b</mi>\n</mrow>\n</math>?</p>","rationale_html":"<p>Choice C is correct. It’s given that the function <math alttext=\"h\"><mi>h</mi>\n</math> is defined by <math alttext=\"h left parenthesis x right parenthesis equals a Superscript x Baseline plus b\"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup><mo>+</mo><mi>b</mi></math> and that the graph of <math alttext=\"y equals h left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo></math> in the <em>xy</em>-plane passes through the points&nbsp;<math alttext=\"left parenthesis 0 comma 10 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mn>10</mn></mrow></mfenced></math> and&nbsp;<math alttext=\"left parenthesis negative 2 comma StartFraction 325 Over 36 EndFraction right parenthesis\"><mfenced><mrow><mo>-</mo><mn>2</mn><mo>,</mo><mo>&nbsp;</mo><mfrac><mn>325</mn><mn>36</mn></mfrac></mrow></mfenced></math>. Substituting <math alttext=\"0\"><mn>0</mn>\n</math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"10\"><mn>10</mn>\n</math> for <math alttext=\"h left parenthesis x right parenthesis\"><mi>h</mi><mfenced><mi>x</mi></mfenced></math> in the equation&nbsp;<math alttext=\"h left parenthesis x right parenthesis equals a Superscript x Baseline plus b\"><mi>h</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup><mo>+</mo><mi>b</mi></math>&nbsp;yields <math alttext=\"10 equals a Superscript 0 Baseline plus b\"><mn>10</mn><mo>=</mo><msup><mi>a</mi><mn>0</mn></msup><mo>+</mo><mi>b</mi></math>, or&nbsp;<math alttext=\"10 equals 1 plus b\"><mn>10</mn><mo>=</mo><mn>1</mn><mo>+</mo><mi>b</mi></math>. Subtracting <math alttext=\"1\"><mn>1</mn>\n</math> from both sides of this equation yields&nbsp;<math alttext=\"9 equals b\"><mn>9</mn><mo>=</mo><mi>b</mi></math>. Substituting&nbsp;<math alttext=\"negative 2\"><mo>-</mo><mn>2</mn></math> for <math alttext=\"x\"><mi>x</mi>\n</math> and <math alttext=\"StartFraction 325 Over 36 EndFraction\"><mfrac><mn>325</mn><mn>36</mn></mfrac></math> for <math alttext=\"h left parenthesis x right parenthesis\"><mi>h</mi><mfenced><mi>x</mi></mfenced></math>&nbsp;in the equation&nbsp;<math alttext=\"h left parenthesis x right parenthesis equals a Superscript x Baseline plus 9\"><mi>h</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup><mo>+</mo><mn>9</mn></math> yields <math alttext=\"StartFraction 325 Over 36 EndFraction equals a Superscript negative 2 Baseline plus 9\"><mfrac><mn>325</mn><mn>36</mn></mfrac><mo>=</mo><msup><mi>a</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mo>+</mo><mn>9</mn></math>. Subtracting <math alttext=\"9\"><mn>9</mn>\n</math> from both sides of this equation yields <math alttext=\"one thirty sixth equals a Superscript negative 2\"><mfrac><mn>1</mn><mn>36</mn></mfrac><mo>=</mo><msup><mi>a</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup></math>, which can be rewritten as <math alttext=\"a squared equals 36\"><msup><mi>a</mi><mn>2</mn></msup><mo>=</mo><mn>36</mn></math>. Taking the square root of both sides of this equation yields&nbsp;<math alttext=\"a equals 6\"><mi>a</mi><mo>=</mo><mn>6</mn></math> and <math alttext=\"a equals negative 6\"><mi>a</mi><mo>=</mo><mo>-</mo><mn>6</mn></math>, but because it’s given that <math alttext=\"a\"><mi>a</mi>\n</math> is a positive constant, <math alttext=\"a\"><mi>a</mi>\n</math> must equal <math alttext=\"6\"><mn>6</mn>\n</math>. Because the value of <math alttext=\"a\"><mi>a</mi>\n</math> is <math alttext=\"6\"><mn>6</mn>\n</math> and the value of <math alttext=\"b\"><mi>b</mi>\n</math> is <math alttext=\"9\"><mn>9</mn>\n</math>, the value of <math alttext=\"a b\"><mrow>\n\t<mi>a</mi>\n\t<mi>b</mi>\n</mrow>\n</math> is <math alttext=\"left parenthesis 6 right parenthesis left parenthesis 9 right parenthesis\"><mo>(</mo><mn>6</mn><mo>)</mo><mo>(</mo><mn>9</mn><mo>)</mo></math>, or <math alttext=\"54\"><mn>54</mn>\n</math>.</p>\n<p><br>Choice A is incorrect and may result from finding the value of <math alttext=\"a Superscript negative 2 Baseline b\"><msup><mi>a</mi><mrow><mo>-</mo><mn>2</mn></mrow></msup><mi>b</mi></math> rather than the value of <math alttext=\"a b\"><mi>a</mi><mi>b</mi></math>.</p>\n<p>Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice D is incorrect and may result from correctly finding the value of <math alttext=\"a\"><mi>a</mi>\n</math> as <math alttext=\"6\"><mn>6</mn>\n</math>, but multiplying it by the <em>y</em>-value in the first ordered pair rather than by the value of <math alttext=\"b\"><mi>b</mi>\n</math>.</p>","correct_answer":["C"],"has_media":false,"has_mathml":true,"has_table":false,"source_created_at":"2023-08-02T20:25:59.827000+00:00","source_updated_at":"2023-08-02T20:25:59.827000+00:00","options":[{"label":"A","content_html":"<p><math><mfrac>\n\t<mn>1</mn>\n\t<mn>4</mn>\n</mfrac>\n</math></p>","ord":0},{"label":"B","content_html":"<p><math><mfrac>\n\t<mn>1</mn>\n\t<mn>2</mn>\n</mfrac>\n</math></p>","ord":1},{"label":"C","content_html":"<p><math><mn>54</mn>\n</math></p>","ord":2},{"label":"D","content_html":"<p><math><mn>60</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}}