Question
Comparable representing a quadrilatic equation has X intercepts at 1, 0 and -3, 0 and a third point at coordinates negative 1/4
Y = -1 (x - 1)(x + 3)
Y = 1(X - 1)(x +3)
Y = -1 (x - 1)( x -3)
Y = -1 (x + 1)(x + 3)
Y = -1 (x - 1)(x + 3)
Y = 1(X - 1)(x +3)
Y = -1 (x - 1)( x -3)
Y = -1 (x + 1)(x + 3)
Answers
GPT-4o mini
Let's analyze the information provided in your question. You mention that there are X-intercepts at \(1\), \(0\), and \(-3\), and a specific point at \((-1/4, Y)\) which you mentioned has coordinates \(Y = -1\), but that could be a typo, as \(Y = 1\) is mentioned too.
However, let's focus on identifying a quadratic function with the given intercepts. Given the X-intercepts at \(1\) and \(-3\), it seems that these correspond to a quadratic polynomial.
From your context, it seems that there might be some confusion about the number of intercepts stated (you provided four potential options). Quadratic functions have at most two X-intercepts. A polynomial having X-intercepts at \(1\) and \(-3\) can be expressed in the following form:
### Formulation from X-Intercepts
Using the X-intercepts provided, the basic form of a quadratic can be written as:
\[
Y = k(x - 1)(x + 3)
\]
where \(k\) is a constant multiplier which can change the vertical stretch/compression and direction of the parabola (upward or downward).
### Evaluating the approach:
Next, we have potential functions provided in the options, namely:
1. \(Y = -1 (x - 1)(x + 3)\)
2. \(Y = 1 (x - 1)(x + 3)\)
3. \(Y = -1 (x - 1)(x - 3)\)
4. \(Y = -1 (x + 1)(x + 3)\)
### Determining correct function:
Since \(1\) and \(-3\) are X-intercepts, the correct function must be formed by the factors \( (x - 1) \) and \( (x + 3) \). Thus we can immediately eliminate options 3 and 4:
We need constant \(k\) to match the point \((-1/4)\) mentioned. Plugging values:
Let's check both valid options:
1. For \(Y = -1 (x - 1)(x + 3)\):
At \(x = -\frac{1}{4}\):
\[
Y = -1 \left(-\frac{1}{4} - 1\right)\left(-\frac{1}{4} + 3\right) = -1 \left(-\frac{5}{4}\right)\left(\frac{11}{4}\right) = -1 \left(-\frac{55}{16}\right) = \frac{55}{16} \neq -1
\]
2. For \(Y = 1 (x - 1)(x + 3)\):
At \(x = -\frac{1}{4}\):
\[
Y = 1 \left(-\frac{1}{4} - 1\right)\left(-\frac{1}{4} + 3\right) = 1 \left(-\frac{5}{4}\right)\left(\frac{11}{4}\right) = -\frac{55}{16} \neq -1
\]
Since neither option yields -1, it appears that we may need a different approach or further clarification on the coordinates you provided.
Given that the correct answer format signals \(Y\) dependent on a constant, without further clarification on the Y-coordinate for point (-1/4), it's not possible to draw a definitive conclusion based on the provided examples.
If you can clarify the specifications or correct any potential errors (i.e., if the Y-coordinate was miswritten), I could help further!
However, let's focus on identifying a quadratic function with the given intercepts. Given the X-intercepts at \(1\) and \(-3\), it seems that these correspond to a quadratic polynomial.
From your context, it seems that there might be some confusion about the number of intercepts stated (you provided four potential options). Quadratic functions have at most two X-intercepts. A polynomial having X-intercepts at \(1\) and \(-3\) can be expressed in the following form:
### Formulation from X-Intercepts
Using the X-intercepts provided, the basic form of a quadratic can be written as:
\[
Y = k(x - 1)(x + 3)
\]
where \(k\) is a constant multiplier which can change the vertical stretch/compression and direction of the parabola (upward or downward).
### Evaluating the approach:
Next, we have potential functions provided in the options, namely:
1. \(Y = -1 (x - 1)(x + 3)\)
2. \(Y = 1 (x - 1)(x + 3)\)
3. \(Y = -1 (x - 1)(x - 3)\)
4. \(Y = -1 (x + 1)(x + 3)\)
### Determining correct function:
Since \(1\) and \(-3\) are X-intercepts, the correct function must be formed by the factors \( (x - 1) \) and \( (x + 3) \). Thus we can immediately eliminate options 3 and 4:
We need constant \(k\) to match the point \((-1/4)\) mentioned. Plugging values:
Let's check both valid options:
1. For \(Y = -1 (x - 1)(x + 3)\):
At \(x = -\frac{1}{4}\):
\[
Y = -1 \left(-\frac{1}{4} - 1\right)\left(-\frac{1}{4} + 3\right) = -1 \left(-\frac{5}{4}\right)\left(\frac{11}{4}\right) = -1 \left(-\frac{55}{16}\right) = \frac{55}{16} \neq -1
\]
2. For \(Y = 1 (x - 1)(x + 3)\):
At \(x = -\frac{1}{4}\):
\[
Y = 1 \left(-\frac{1}{4} - 1\right)\left(-\frac{1}{4} + 3\right) = 1 \left(-\frac{5}{4}\right)\left(\frac{11}{4}\right) = -\frac{55}{16} \neq -1
\]
Since neither option yields -1, it appears that we may need a different approach or further clarification on the coordinates you provided.
Given that the correct answer format signals \(Y\) dependent on a constant, without further clarification on the Y-coordinate for point (-1/4), it's not possible to draw a definitive conclusion based on the provided examples.
If you can clarify the specifications or correct any potential errors (i.e., if the Y-coordinate was miswritten), I could help further!