{"id":"cb_question_bank:68972dc6-a2d0-4672-a66a-fdc754fb35db","source_id":"cb_question_bank","source_item_id":"68972dc6-a2d0-4672-a66a-fdc754fb35db","external_id":"e0e29be6-463f-477b-a2c1-4960db142fa1","ibn":null,"question_id":"f2d60b99","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":"The function f left parenthesis x right parenthesis equals one ninth left parenthesis x minus 7 right parenthesis squared plus 3 gives a metal ball's height above the ground f left parenthesis x right parenthesis , in inches, x seconds after it started moving on a track, where 0 less than or equals x less than or equals 10 . Which of the following is the best interpretation of the vertex of the graph of y equals f left parenthesis x right parenthesis in the xy -plane?","stimulus":"","stimulus_html":null,"stem_html":"<p style=\"text-align: left;\">The function <math alttext=\"f left parenthesis x right parenthesis equals one ninth left parenthesis x minus 7 right parenthesis squared plus 3\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>9</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>7</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>3</mn></math> gives a metal ball's height above the ground <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math>, in inches, <math alttext=\"x\"><mi>x</mi>\n</math> seconds after it started moving on a track, where <math alttext=\"0 less than or equals x less than or equals 10\"><mn>0</mn><mo>≤</mo><mi>x</mi><mo>≤</mo><mn>10</mn></math>. Which of the following is the best interpretation of the vertex of 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>&nbsp;in the <em>xy</em>-plane?</p>","rationale_html":"<p style=\"text-align: left;\">Choice A is correct. The graph of a quadratic equation in the form <math alttext=\"y equals a left parenthesis x minus h right parenthesis squared plus k\"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mi>k</mi></math>, where <math alttext=\"a\"><mi>a</mi>\n</math>, <math alttext=\"h\"><mi>h</mi>\n</math>, and <math alttext=\"k\"><mi>k</mi>\n</math> are positive constants, is a parabola that opens upward with vertex <math alttext=\"left parenthesis h comma k right parenthesis\"><mfenced><mrow><mi>h</mi><mo>,</mo><mi>k</mi></mrow></mfenced></math>. The given function <math alttext=\"f left parenthesis x right parenthesis equals one ninth left parenthesis x minus 7 right parenthesis squared plus 3\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>9</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>7</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>3</mn></math>&nbsp;is in the form <math alttext=\"y equals a left parenthesis x minus h right parenthesis squared plus k\"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mi>h</mi></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mi>k</mi></math>, where <math alttext=\"y equals f left parenthesis x right parenthesis\"><mi>y</mi><mo>=</mo><mi>f</mi><mfenced><mi>x</mi></mfenced></math>, <math alttext=\"a equals one ninth\"><mi>a</mi><mo>=</mo><mfrac><mn>1</mn><mn>9</mn></mfrac></math>, <math alttext=\"h equals 7\"><mrow>\n\t<mi>h</mi>\n\t<mo>=</mo>\n\t<mn>7</mn>\n</mrow>\n</math>, and <math alttext=\"k equals 3\"><mrow>\n\t<mi>k</mi>\n\t<mo>=</mo>\n\t<mn>3</mn>\n</mrow>\n</math>. Therefore, 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> is a parabola that opens upward with vertex <math alttext=\"left parenthesis 7 comma 3 right parenthesis\"><mfenced><mrow><mn>7</mn><mo>,</mo><mn>3</mn></mrow></mfenced></math>. Since the parabola opens upward, the vertex is the lowest point on the graph. It follows that the <em>y</em>-coordinate of the vertex of 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> is the minimum value of <math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math>. Therefore, the minimum value of&nbsp;<math alttext=\"f left parenthesis x right parenthesis\"><mi>f</mi><mfenced><mi>x</mi></mfenced></math> is <math alttext=\"3\"><mn>3</mn>\n</math>. It’s given that <math alttext=\"f left parenthesis x right parenthesis equals one ninth left parenthesis x minus 7 right parenthesis squared plus 3\"><mi>f</mi><mfenced><mi>x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mn>9</mn></mfrac><msup><mfenced><mrow><mi>x</mi><mo>-</mo><mn>7</mn></mrow></mfenced><mn>2</mn></msup><mo>+</mo><mn>3</mn></math> represents the metal ball’s height above the ground, in inches, <math alttext=\"x\"><mi>x</mi>\n</math> seconds after it started moving on a track. Therefore, the best interpretation of the vertex of 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> is that the metal ball’s minimum height was <math alttext=\"3\"><mn>3</mn>\n</math> inches above the ground.</p>\n<p>Choice B is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice C is incorrect and may result from conceptual or calculation errors.</p>\n<p>Choice D is incorrect and may result from conceptual or calculation errors.</p>","correct_answer":["A"],"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 metal ball's minimum height was <math alttext=\"3\"><mn>3</mn>\n</math> inches above the ground.</p>","ord":0},{"label":"B","content_html":"<p style=\"text-align: left;\">The metal ball's minimum height was <math alttext=\"7\"><mn>7</mn>\n</math> inches above the ground.</p>","ord":1},{"label":"C","content_html":"<p style=\"text-align: left;\">The metal ball's height was <math alttext=\"3\"><mn>3</mn>\n</math> inches above the ground when it started moving.</p>","ord":2},{"label":"D","content_html":"<p style=\"text-align: left;\">The metal ball's height was <math alttext=\"7\"><mn>7</mn>\n</math> inches above the ground when it started moving.</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}}