In the xy-plane, a parabola has vertex (9, −14) and intersects the x-axis at two points. If the equation of the parabola is written in the form y = ax^2 + bx+ c, where a, b, and c are constants, which of the following could be the value of a+b+c?
","rationale_html":"Choice D is correct. The equation of a parabola in the xy-plane can be written in =( -)2 + ( ) the form y a x h k, where a is a constant and h, k is the vertex of the parabola. If a is positive, the parabola will open upward, and if a is negative, the ( - ) parabola will open downward. It’s given that the parabola has vertex 9, 14 . - =( -)2 + Substituting 9 for h and 14 for k in the equation y a x h k gives =( )-2 - =( -)( -) - y a x 9 14, which can be rewritten as y a x 9 x 9 14, or (=2 - +) - y a x 18 x 81 14. Distributing the factor of a on the right-hand side of =2 - + - this equation yields y ax 18ax 81a 14. Therefore, the equation of the =2 - + - =2 + + parabola, y ax 18ax 81a 14, can be written in the form y ax bx c, = =- = - - - where a a, 1b8 , and a 8c1 14a. Substituting 18a for b and 81a 14 + + ( ) +( - ) +( - ) - for c in the expression a b c yields a18 8a1 14a, or 64a 14. ( - ) Since the vertex of the parabola, 9, 14 , is below the x-axis, and it’s given that the parabola intersects the x-axis at two points, the parabola must open upward. Therefore, the constant a must have a positive value. Setting the expression - - =- 64a 14 equal to the value in choice D yields 64a 14 12. Adding 14 to both = sides of this equation yields 64a 2. Dividing both sides of this equation by 64 = 2 yields a , which is a positive value. Therefore, if the equation of the parabola is 64 =2 + + written in the form y ax bx c, where a, b, and c are constants, the value of + + - a b c could be 12. ( - ) Choice A is incorrect. If the equation of a parabola with a vertex at 9, 14 =2 + + is written in the form y ax bx c, where a, b, and c are constants and + + =- a b c 23, then the value of a will be negative, which means the parabola will open downward, not upward, and will intersect the x-axis at zero points, not two points. Choice B is incorrect. If the equation of a parabola with a vertex at ( - ) =2 + + 9, 14 is written in the form y ax bx c, where a, b, and c are constants + + =- and a b c 19, then the value of a will be negative, which means the parabola will open downward, not upward, and will intersect the x-axis at zero points, not two points. Choice C is incorrect. If the equation of a parabola with a ( - ) =2 + + vertex at 9, 14 is written in the form y ax bx c, where a, b, and c are + + =- constants and a b c 14, then the value of a will be 0, which is inconsistent with the equation of a parabola. 42 SAT PRACTICE TEST #4 ANSWER EXPLANATIONS SAT ANSWER EXPLANATIONS nnn MATH: MODULE 1
","correct_answer":["D"],"has_media":false,"has_mathml":false,"has_table":false,"source_created_at":null,"source_updated_at":null,"options":[{"label":"A","content_html":"−23
","ord":0},{"label":"B","content_html":"−19
","ord":1},{"label":"C","content_html":"−14
","ord":2},{"label":"D","content_html":"−12
","ord":3}],"attribution":{"source_id":"cb_practice_pdf","source_name":"College Board full-length SAT practice tests (PDF)","rights_holder":"College Board","canonical_url":"https://satsuite.collegeboard.org/practice/practice-tests/paper","retrieved_from":"https://satsuite.collegeboard.org/practice/practice-tests/paper","attribution":"Full-length SAT practice tests 4–11 © College Board, published free at satsuite.collegeboard.org.","disclaimer":"Extracted from PDF layout: equations, figures and tables do not survive intact. Use the original PDFs for anything the text renders poorly. SAT® is a trademark registered by the College Board, which is not affiliated with and does not endorse this project.","license":"College Board copyright. Free to download; not licensed for redistribution.","redistributable":false,"item_count":905}}