Question
A parable representing a quadratic equation has x-intercepts and 1, 0 and 3, 0 and a third point with coordinates at 4, 3 which quadratic equation in an intercept from accurately represents this information
Y = 1 x - 1x + 3 y = 1 X - 1 x - 3 = -1 x - 1 x - 3 y = 1x + 1X - 3
Y = 1 x - 1x + 3 y = 1 X - 1 x - 3 = -1 x - 1 x - 3 y = 1x + 1X - 3
Answers
GPT-4o mini
To represent the quadratic equation with the x-intercepts given as (1, 0) and (3, 0), we first need to write the quadratic equation in factored form. The roots (or x-intercepts) give us the factors of the equation.
The quadratic equation can be represented as:
\[
y = a(x - 1)(x - 3)
\]
where \( a \) is a constant.
Next, we can use the additional point (4, 3) to determine the value of \( a \).
Plugging \( x = 4 \) and \( y = 3 \) into the equation:
\[
3 = a(4 - 1)(4 - 3)
\]
This simplifies to:
\[
3 = a(3)(1)
\]
\[
3 = 3a
\]
Now, solve for \( a \):
\[
a = 1
\]
Thus, the quadratic equation becomes:
\[
y = 1(x - 1)(x - 3) = (x - 1)(x - 3)
\]
Expanding this expression gives:
\[
y = x^2 - 4x + 3
\]
Therefore, the quadratic equation that accurately represents the information provided, with x-intercepts at (1, 0) and (3, 0), and passing through the point (4, 3), is:
\[
y = x^2 - 4x + 3
\]
The quadratic equation can be represented as:
\[
y = a(x - 1)(x - 3)
\]
where \( a \) is a constant.
Next, we can use the additional point (4, 3) to determine the value of \( a \).
Plugging \( x = 4 \) and \( y = 3 \) into the equation:
\[
3 = a(4 - 1)(4 - 3)
\]
This simplifies to:
\[
3 = a(3)(1)
\]
\[
3 = 3a
\]
Now, solve for \( a \):
\[
a = 1
\]
Thus, the quadratic equation becomes:
\[
y = 1(x - 1)(x - 3) = (x - 1)(x - 3)
\]
Expanding this expression gives:
\[
y = x^2 - 4x + 3
\]
Therefore, the quadratic equation that accurately represents the information provided, with x-intercepts at (1, 0) and (3, 0), and passing through the point (4, 3), is:
\[
y = x^2 - 4x + 3
\]