Final answer:
The algebraic representation of the given sequence is f(n) = n^2, which is already a quadratic pattern. The sequence equals 120 at the 11th term, and the sequence never equals -1 as there is no real number that squares to -1. To graph the parabola, plot points using n as x-values and the sequence terms as y-values, then draw the curve.
Step-by-step explanation:
To solve the student's question regarding the sequence 0,3,8,15,24,... where 0 is the 0th term, we need to find a pattern and establish an algebraic representation for both the linear and quadratic patterns in the sequence.
Algebraic Representation for the Linear Pattern
The difference between consecutive terms increases by 2 each time (3, 5, 7, 9, ...). This resembles the odd numbers sequence, and odd numbers can be represented as 2n+1, where n starts from 0. So, the nth term for the linear pattern can be written as the sum of the first n odd numbers, which is equal to n^2. Therefore, the given sequence in terms of n can be f(n) = n^2.
Finding when the Sequence Equals 120
To find when the sequence equals 120, we solve the equation n^2 = 120. Finding the square root of 120 gives us two solutions, n = 10.95 or n = -10.95. Since n represents the position in the sequence and cannot be negative, the sequence is equal to 120 at the 11th term (considering we start counting from n = 0).
Finding when the Sequence Equals -1
To find when the sequence equals -1, we solve the equation n^2 = -1. However, there's no real number that can be squared to give -1, so there is no term in the sequence that will equal -1.
Algebraic Representation for the Quadratic Sequence
The sequence itself is already quadratic, with the algebraic representation being f(n) = n^2.
Graphing the Parabola
To graph the parabola, plot points with the x-values as the position in the sequence (n) and the y-values as the corresponding term in the sequence. Then draw a curve that passes through these points, showing the shape of y = n^2.