{"id":"cb_question_bank:21ccba60-03eb-46d4-8fdb-dd3558e8f879","source_id":"cb_question_bank","source_item_id":"21ccba60-03eb-46d4-8fdb-dd3558e8f879","external_id":"35b45e44-2494-439f-a174-5c714299c55c","ibn":null,"question_id":"4a0d0399","program":"SAT","module":"math","domain_code":"P","domain":"Advanced Math","skill_code":"P.C.","skill":"Nonlinear functions","difficulty":"H","score_band":7,"answer_type":"spr","answer_source":"key","layout_warning":null,"form_id":null,"module_number":null,"position":null,"stem":"The function f is defined by f left parenthesis x right parenthesis equals a Superscript x Baseline plus b , where a and b are constants. In the x y -plane, the graph of y equals f left parenthesis x right parenthesis has an x -intercept at left parenthesis 2 comma 0 right parenthesis and a y -intercept at left parenthesis 0 comma negative 323 right parenthesis . What is the value of b ?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The function <math alttext=\"f\"><mi>f</mi>\n</math> is defined by&nbsp;<math alttext=\"f left parenthesis x right parenthesis equals a Superscript x Baseline plus b\"><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup><mo>+</mo><mi>b</mi></math>, where <math alttext=\"a\"><mi>a</mi>\n</math> and <math alttext=\"b\"><mi>b</mi>\n</math> are constants. In the <math alttext=\"x y\"><mrow>\n\t<mi>x</mi>\n\t<mi>y</mi>\n</mrow>\n</math>-plane, the graph of <math alttext=\"y equals f left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> has an <math alttext=\"x\"><mi>x</mi>\n</math>-intercept at&nbsp;<math alttext=\"left parenthesis 2 comma 0 right parenthesis\"><mfenced><mrow><mn>2</mn><mo>,</mo><mn>0</mn></mrow></mfenced></math> and a <math alttext=\"y\"><mi>y</mi>\n</math>-intercept at <math alttext=\"left parenthesis 0 comma negative 323 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mrow><mo>-</mo><mn>323</mn></mrow></mrow></mfenced></math>. What is the value of <math alttext=\"b\"><mi>b</mi>\n</math>?</p>","rationale_html":"<p style=\"text-align: left;\">The correct answer is <math alttext=\"negative 324\"><mo>-</mo><mn>324</mn></math>. It's given that the function&nbsp;<math alttext=\"f\"><mi>f</mi></math> is defined by&nbsp;<math alttext=\"f left parenthesis x right parenthesis equals a Superscript x Baseline plus b\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup><mo>+</mo><mi>b</mi></math>, where&nbsp;<math alttext=\"a\"><mi>a</mi></math> and&nbsp;<math alttext=\"b\"><mi>b</mi></math> are constants. It's also given that the graph of&nbsp;<math alttext=\"y equals f left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math> has a&nbsp;<em>y</em>-intercept at&nbsp;<math alttext=\"left parenthesis 0 comma negative 323 right parenthesis\"><mfenced><mrow><mn>0</mn><mo>,</mo><mo>-</mo><mn>323</mn></mrow></mfenced></math>. It follows that&nbsp;<math alttext=\"f left parenthesis 0 right parenthesis equals negative 323\"><mi>f</mi><mfenced><mn>0</mn></mfenced><mo>=</mo><mo>-</mo><mn>323</mn></math>. Substituting&nbsp;<math alttext=\"0\"><mn>0</mn></math> for&nbsp;<math alttext=\"x\"><mi>x</mi></math> and&nbsp;<math alttext=\"negative 323\"><mo>-</mo><mn>323</mn></math> for&nbsp;<math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> in&nbsp;<math alttext=\"f left parenthesis x right parenthesis equals a Superscript x Baseline plus b\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><msup><mi>a</mi><mi>x</mi></msup><mo>+</mo><mi>b</mi></math> yields&nbsp;<math alttext=\"negative 323 equals a Superscript 0 Baseline plus b\"><mo>-</mo><mn>323</mn><mo>=</mo><msup><mi>a</mi><mn>0</mn></msup><mo>+</mo><mi>b</mi></math>, or&nbsp;<math alttext=\"negative 323 equals 1 plus b\"><mo>-</mo><mn>323</mn><mo>=</mo><mn>1</mn><mo>+</mo><mi>b</mi></math>. Subtracting&nbsp;<math alttext=\"1\"><mn>1</mn></math> from each side of this equation yields&nbsp;<math alttext=\"negative 324 equals b\"><mo>-</mo><mn>324</mn><mo>=</mo><mi>b</mi></math>. Therefore, the value of&nbsp;<math alttext=\"b\"><mi>b</mi></math> is&nbsp;<math alttext=\"negative 324\"><mo>-</mo><mn>324</mn></math>.</p>","correct_answer":["-324"],"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":[],"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}}