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ok this is a very confusing topic. im doing the review for my midterm and encountered a problem, heres the question:
g(x)=x^2-2x+5 professors answer =vertex is (1,4) and x-intercept is none and y--intercept is (0,5)
the only thing i got right was the y-intercept. for the vertex im getting (1,6), and why is there not an x-intercept?
id appreciate any help? thank u!
You got the x part of the vertex correct. To find the y coordinate, plug x=1 back in. What's g(1)?
You find the x-intercept by setting g(x)=0 and solving. Can you solve it? What happens when you use the quadratic formula to find the x-intercepts. Also, you may notice that the coefficient on theis positive, which means the parabola opens upwards, and the vertex is above the x-axis (4 is a positive number). Since the vertex is the lowest point of the parabola and it only goes up from there, it can never cross the x-axis.
Quote:
Originally Posted by noork85
hence :
therefore :
for vertex we have :
Sincegraph of the function and x-axis have no common points .
can i use this formula for all that are in the format of g(x)=ax^2+bx+c?
Quote:
Originally Posted by princeps
hence :
therefore :
for vertex we have :
Since
and im still confused about the x-intercept. i know im slow....but ...please help!
and whats the D stand for? i know is vertex?
still lost, where does the -16 come from? can u break down step by tiny step?
D is the discriminant. If you haven't learned it yet, when D>0, then there are two roots. When D=0, there is one root. When D<0, then there are no roots.
Notice that the discriminant is the part inside of the square root for the quadratic formula.As princeps pointed out,
is a negative number. You can't take a square root of a negative number and get a real number for a result. In other words, you can't square any real number and get -16.
Step by step:
You know what a, b, and c are, so just substitute and do the arithmetic.
Alternately, if you had gone straight to the quadratic formula to find the x-intercepts, you would get:
The quadratic formula will always give you the x-intercepts for a quadratic function (if they exist), but you can't find the square root of -16, so there are no real number x-intercepts.
oh thank u so much! that really cleared it up. i didnt learn anything about discriminitives. but yes, i shouldve just gone straight to the quadratic formula to solve it.
can u also please explain the vertex part.
It might be helpful if you showed how you got to your answer.
You found the x value for the vertex already. x=1 is correct. Let's split the denominator of the quadratic formula so that it's. Notice that you add or subtract
from
. Also remember that the vertex of a parabola is on it's axis of symmetry. So if you add or subtract the same amount from a number to get the x-intercepts, then the axis of symmetry is at
.
Plug in the numbers and do the arithmetic and you get x=1, which you already knew. At this point, noork85 went a seperate route to find the y value, which I'm not familiar with, so here's the way that I do it, which seems more intuitive:
Where is the parabola at when x=1? It's at g(1). All you have to do is go back to your original function and plug 1 in for x and do the arithmetic.
thank u so very much! very detailed and helpful!