Questions and Answers
Which of the following could be the graph of $y=x^2 -2$?
Which equation could be solved using the graph above?
What is the solution to $3x^2 +3x+5=0$?
Option D
What are the zeroes of $y=x^2 +2x-8$?
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How many real solutions does the function shown on the graph above have?
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At what time will the football be 25 feet above ground, given the equation $s(t)=-16t^2 +50t+4$?
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What are the dimensions of a rectangular window with an area of 2,720 square inches and a side length ratio of 1.7 to 1?
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What is the solution to the equation $x^2 +14x+48=0$?
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What should Max charge per poster to make maximum profit, and what is the maximum profit he can make in a month based on the function $p(x)=-10x^2 +200x-250$?
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Use the graphing to find the solutions to the system of equations $x^2 -y=4$ and $2x+y=-1$.
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Which of the following are the most likely factors of the function graphed above?
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Write the equation of the parabola in vertex form with an axis of symmetry at $x=-8$, a maximum height of 2, and passing through the point (-7,-1).
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Any number in the form of $a+bi$, where $a$ and $b$ are real numbers and $b$ doesn't equal 0 is considered a pure imaginary number.
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The minimum value of a function is the smallest y-value of the function.
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$i^2=$ the square root of -1.
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Complex numbers can be graphed on the real xy coordinate plane.
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A quadratic equation can be written in vertex form or in standard. Sometimes one form is more beneficial than the other. Identify which form would be more helpful if you needed to do each task listed below and explain why.
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Study Notes
Quadratic Functions and Equations Study Notes
- The graph of ( y = x^2 - 2 ) represents a vertical shift of the graph ( y = x^2 ) downwards by 2 units.
- Certain quadratic equations can be solved graphically by identifying intersections, such as ( y = x^2 - y = 4 ) and ( 2x + y = -1 ).
- The equation ( 3x^2 + 3x + 5 = 0 ) has no real solutions due to a negative discriminant.
- The zeroes of the quadratic ( y = x^2 + 2x - 8 ) can be found using factoring, resulting in roots at ( x = -4 ) and ( x = 2 ).
- A quadratic function can have zero, one, or two real solutions determined by the discriminant; the example graph indicates two real solutions.
Applications of Quadratics
- The height of a football kicked into the air can be modeled by the function ( s(t) = -16t^2 + 50t + 4 ); it reaches 25 feet at ( t = 0.5 ) seconds and ( t = 2.625 ) seconds.
- A rectangular window with a side ratio of 1.7:1 and an area of 2,720 square inches has dimensions of 40 inches by 68 inches.
- To maximize profit selling posters, Max should charge $10 per poster, resulting in a maximum profit of $750, as described by the profit function ( p(x) = -10x^2 + 200x - 250 ).
Solving Quadratic Equations
- The solutions to the equation ( x^2 + 14x + 48 = 0 ) extract roots of ( x = -6 ) or ( x = -8 ).
- Factors of the quadratic function found through graphing may include ( (x + 3)(x - 4) ), which match its zeroes.
Parabola Characteristics
- Parabolas can be expressed in vertex form ( y = a(x-h)^2 + k ), where ( (h, k) ) is the vertex; for a parabola with an axis of symmetry at ( x = -8 ) and a vertex at height 2, the equation would be ( y = -3(x + 8)^2 + 2 ).
Complex Numbers
- Pure imaginary numbers are in the form ( a + bi ) only when ( b \neq 0 ) and ( a = 0 ).
- Instant calculations like ( i^2 ) equal to -1 reinforce the foundational properties of imaginary units.
- Complex numbers are not represented on the traditional ( xy ) coordinate system, as they exist in a different dimensional space.
Truth Statements
- The minimum value of a quadratic function refers to the smallest ( y )-value attained by the function, confirming its defining characteristic.
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Description
This quiz covers key concepts related to quadratic functions and equations, including their graphs, solutions, and applications. Learn to identify characteristics such as vertical shifts, intersections, and real solutions using the quadratic formula and factoring methods.
