Quadratic: how do I get the vertex from y = 3x^2 - 24x + 17?

[megan0430]

The problem is a word problem and i ended up getting this equation.

. . .y = 3x^2 - 24x + 17

This is a standard quadratic equation. I understand that the y-intercept is at y = 17, or the point (0, 17), but I need to know the vertex in order to graph it.

HOW do I get the vertex from here?
Please help. Thank you!

 

[tkhunny]

Re: Quadratic: how do I get the vertex from y = 3x^2 - 24x +

y = 3x^2 - 24x + 17

Completing the Square?

y = 3x^2 - 24x + 17

y = (3x^2 - 24x) + 17

y = 3(x^2 - 8x) + 17

Compute Magic Number
-8/2 = -4
(-4)^2 = 16

y = 3(x^2 - 8x + 16) + 17 - 3(16) <== This may be the tricky step. Don't forget the Distributive Property

y = 3(x - 4)^2 - 31

Looks like (4,-31) to me.

 

[arthur ohlsten]

the equation is that of a conic section. With a first order y ,it is the equation of a parabola. The standard equation would be y=k[x-a]^2+b
If k is + the parabola is open up ,if k is -it is open down. Vertex at a,b

y=3x^2-24x+17 rewrite grouping the x terms
y=[3x^2-24x ] +17 factor out 3
y=3[x^2-8x ]+17 complete the square by adding and
subtracting 1/2 the x coefficient squared
y=3[x^2-8x+16]-48+17
y=3[x-4]^2-31 a parabola open up vertex at 4,-31

 

[Mrspi]

Re: Quadratic: how do I get the vertex from y = 3x^2 - 24x +

The problem is a word problem and i ended up getting this equation.

. . .y = 3x^2 - 24x + 17

This is a standard quadratic equation. I understand that the y-intercept is at y = 17, or the point (0, 17), but I need to know the vertex in order to graph it.

HOW do I get the vertex from here?
Please help. Thank you!

You've been shown how to complete the square.

I'll offer an alternative approach.

For the quadratic function

y = ax<SUP>2</SUP> + bx + c

the vertex occurs when x = -b / 2a

In your equation, a = 3 and b = -24. So, the vertex occurs when

x = -(-24) / 2*3
x = 24/6
x = 4

To find the y-coordinate for the vertex, substitute 4 for x in the equation:
y = 3x<SUP>2</SUP> - 24x + 17
y = 3(4)<SUP>2</SUP> - 24(4) + 17
y = 3(16) - 96 + 17
y = 48 - 96 + 17
y = -31

And, as you've already seen by the method of completing the square, the vertex is at (4, -31).

I think that knowing more than one way to approach a problem can't hurt.