How do you write a quadratic equation with y-intercept of -4 and vertex at (3, -7)?

Question

Isaiah Anthony · Accepted Answer

#y=1/3x^2-2x-4# The equation of a parabola in #color(blue)"vertex form"# is. #color(red)(bar(ul(|color(white)(2/2)color(black)(y=a(x-h)^2+k)color(white)(2/2)|)))# where (h , k) are the coordinates of the vertex and a is a constant. #"here " (h,k)=(3,-7)# #rArry=a(x-3)^2-7# #"to find a, use y-intercept of - 4"rarr(0,-4)# #-4=9a-7# #rArra=1/3# #rArry=1/3(x-3)^2-7larrcolor(red)" in vertex form"# #"distributing and simplifying gives"# #y=1/3(x^2-6x+9)-7# #rArry=1/3x^2-2x-4larrcolor(red)" in standard form"#

Nathan Axel · Answer

undefined To write a quadratic equation with a y-intercept of -4 and a vertex at (3, -7), you can use the vertex form of a quadratic equation, which is $ y = a(x - h)^2 + k $. The vertex form allows you to easily determine the vertex of the parabola, where (h, k) represents the vertex.

Given that the vertex is (3, -7), substitute h = 3 and k = -7 into the vertex form equation:

$$ y = a(x - 3)^2 - 7 $$

Now, to find the value of 'a', you can use the y-intercept (-4). Plug in x = 0 and y = -4 into the equation:

$$ -4 = a(0 - 3)^2 - 7 $$

Solve for 'a':

$$ -4 = 9a - 7 $$
$$ 3 = 9a $$
$$ a = \frac{1}{3} $$

So, the quadratic equation is:

$$ y = \frac{1}{3}(x - 3)^2 - 7 $$

Eva Abramson · Answer

undefined To write a quadratic equation with a y-intercept of -4 and a vertex at (3, -7), we can use the vertex form of a quadratic equation, which is:

$$ y = a(x - h)^2 + k $$

Where (h, k) is the vertex of the parabola.

Given that the vertex is (3, -7), we have:
$$ h = 3 $$ 
$$ k = -7 $$

We also know that the y-intercept is at (0, -4), so when x = 0, y = -4. Substituting these values into the equation gives us:

$$ -4 = a(0 - 3)^2 - 7 $$

Solving this equation for 'a':

$$ -4 = a(-3)^2 - 7 $$
$$ -4 = 9a - 7 $$
$$ 3 = 9a $$
$$ a = \frac{3}{9} $$
$$ a = \frac{1}{3} $$

Now that we have 'a', 'h', and 'k', we can write the equation:

$$ y = \frac{1}{3}(x - 3)^2 - 7 $$

Therefore, the quadratic equation with a y-intercept of -4 and a vertex at (3, -7) is:
$$ y = \frac{1}{3}(x - 3)^2 - 7 $$