Question
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Part 1
Write an equation for a parabola with x-intercepts left parenthesis negative 2 comma 0 right parenthesis(−2,0) and left parenthesis 4 comma 0 right parenthesis(4,0) which passes through the point left parenthesis 2 comma negative 40 right parenthesis(2,−40).
Question content area bottom
Part 1
Write the equation.
Part 1
Write an equation for a parabola with x-intercepts left parenthesis negative 2 comma 0 right parenthesis(−2,0) and left parenthesis 4 comma 0 right parenthesis(4,0) which passes through the point left parenthesis 2 comma negative 40 right parenthesis(2,−40).
Question content area bottom
Part 1
Write the equation.
Answers
Answered by
GPT-4o mini
To write the equation of a parabola given the x-intercepts and a point it passes through, we can use the factored form of a quadratic equation.
Given the x-intercepts at \( (-2, 0) \) and \( (4, 0) \), we can express the quadratic equation as:
\[
y = a(x + 2)(x - 4)
\]
where \( a \) is a constant that will determine the vertical stretch or compression of the parabola.
Next, we are given that the parabola passes through the point \( (2, -40) \). We can use this point to find the value of \( a \). By substituting \( x = 2 \) and \( y = -40 \) into the equation, we have:
\[
-40 = a(2 + 2)(2 - 4)
\]
Calculating the values inside the parentheses:
\[
-40 = a(4)(-2)
\]
This simplifies to:
\[
-40 = -8a
\]
Now we can solve for \( a \):
\[
a = \frac{-40}{-8} = 5
\]
Now that we have the value of \( a \), we can substitute it back into the equation:
\[
y = 5(x + 2)(x - 4)
\]
Next, we can expand this expression:
\[
y = 5[(x + 2)(x - 4)]
\]
Using the distributive property to expand:
\[
y = 5[x^2 - 4x + 2x - 8]
\]
\[
y = 5[x^2 - 2x - 8]
\]
\[
y = 5x^2 - 10x - 40
\]
Thus, the equation of the parabola is:
\[
y = 5x^2 - 10x - 40
\]
Given the x-intercepts at \( (-2, 0) \) and \( (4, 0) \), we can express the quadratic equation as:
\[
y = a(x + 2)(x - 4)
\]
where \( a \) is a constant that will determine the vertical stretch or compression of the parabola.
Next, we are given that the parabola passes through the point \( (2, -40) \). We can use this point to find the value of \( a \). By substituting \( x = 2 \) and \( y = -40 \) into the equation, we have:
\[
-40 = a(2 + 2)(2 - 4)
\]
Calculating the values inside the parentheses:
\[
-40 = a(4)(-2)
\]
This simplifies to:
\[
-40 = -8a
\]
Now we can solve for \( a \):
\[
a = \frac{-40}{-8} = 5
\]
Now that we have the value of \( a \), we can substitute it back into the equation:
\[
y = 5(x + 2)(x - 4)
\]
Next, we can expand this expression:
\[
y = 5[(x + 2)(x - 4)]
\]
Using the distributive property to expand:
\[
y = 5[x^2 - 4x + 2x - 8]
\]
\[
y = 5[x^2 - 2x - 8]
\]
\[
y = 5x^2 - 10x - 40
\]
Thus, the equation of the parabola is:
\[
y = 5x^2 - 10x - 40
\]