Question

A parable representing a quadratic equation has x-intercepts and one, 0 and 3, 0 and a third point with coordinates at 4, 3 which quadratic equation in an intercept from accurately represents this information

Answer 1

To find a quadratic equation that has x-intercepts at \( (0, 0) \) and \( (3, 0) \), and passes through the point \( (4, 3) \), we start with the general form of a quadratic equation based on its roots.

Since the x-intercepts are at \( 0 \) and \( 3 \), we can express the quadratic in factored form as:

\[
y = k(x)(x - 3)
\]

where \( k \) is a constant that determines the vertical stretch or compression.

Next, we need to find the value of \( k \). We know that the parabola passes through the point \( (4, 3) \). So, we can substitute \( x = 4 \) and \( y = 3 \) into the equation to solve for \( k \):

\[
3 = k(4)(4 - 3)
\]
\[
3 = k(4)(1)
\]
\[
3 = 4k
\]

Solving for \( k \):

\[
k = \frac{3}{4}
\]

Now, we can plug the value of \( k \) back into the equation:

\[
y = \frac{3}{4}(x)(x - 3)
\]

Expanding this equation:

\[
y = \frac{3}{4}(x^2 - 3x)
\]
\[
y = \frac{3}{4}x^2 - \frac{9}{4}x
\]

Thus, the quadratic equation that represents the given information is:

\[
y = \frac{3}{4}x^2 - \frac{9}{4}x
\]

This equation has x-intercepts at \( (0, 0) \) and \( (3, 0) \), and it passes through the point \( (4, 3) \).