The 1st thing to note about this equation is that there are no values of x that will make g not a function. This means that the sketch of this equation is going to be smooth and curvy without any breaks. Also, go ahead and draw your coordinate axis.
The y intercept is at y = 24. Go ahead and put a dot there.
Next step: Factor the equation into g(x) = (x - 6)(-x - 4).
From this you can see g = 0 at x = 6 and x = -4. Go ahead an put dots on the x axis at 6 and -4.
The next thing you need to do is take the 1st derivative of g.
g'(x) = -2x + 2
The derivative is equal to 0 at x = 1. This means the slope of the line changes from being + to - as we cross x = 1. Go ahead and put a vertical dashed line at x = 1. At this point, I would go ahead and sketch the curve. It looks like an inverted parabola with it's center about x = 1 and crossing the x axis at 4 and 6 and the y axis at 24.
X^2-2x-24=0 What 2 numbers are you able to come back up with that stick to those rules? -whilst extra at the same time, they equivalent -2 -whilst expanded at the same time, they equivalent -24 2 numbers that extra healthful are 4 and -6. So the factored style could be (X-4)(X+6)
The coordinates for the x-intercepts are (6,0) and (-4,0) because the x-intercepts are where the curve hits the x-axis, AKA where g(x)=0.
To sketch, first draw these points on the coordinate plane. Because the x^2 term has a negative in front of it, the graph will open down like a frown (yes it rhymes), so connect the points (6,0) and (-4,0) with a downward facing parabola.
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The 1st thing to note about this equation is that there are no values of x that will make g not a function. This means that the sketch of this equation is going to be smooth and curvy without any breaks. Also, go ahead and draw your coordinate axis.
The y intercept is at y = 24. Go ahead and put a dot there.
Next step: Factor the equation into g(x) = (x - 6)(-x - 4).
From this you can see g = 0 at x = 6 and x = -4. Go ahead an put dots on the x axis at 6 and -4.
The next thing you need to do is take the 1st derivative of g.
g'(x) = -2x + 2
The derivative is equal to 0 at x = 1. This means the slope of the line changes from being + to - as we cross x = 1. Go ahead and put a vertical dashed line at x = 1. At this point, I would go ahead and sketch the curve. It looks like an inverted parabola with it's center about x = 1 and crossing the x axis at 4 and 6 and the y axis at 24.
X^2-2x-24=0 What 2 numbers are you able to come back up with that stick to those rules? -whilst extra at the same time, they equivalent -2 -whilst expanded at the same time, they equivalent -24 2 numbers that extra healthful are 4 and -6. So the factored style could be (X-4)(X+6)
First, factor the equation.
g(x)=(6-x)(4+x)
To find x-intercepts, set g(x)=0.
0=(6-x)(4+x)
Use the zero product property.
0=6-x and 0=4+x
Solve for x
x=6 and x= -4
The coordinates for the x-intercepts are (6,0) and (-4,0) because the x-intercepts are where the curve hits the x-axis, AKA where g(x)=0.
To sketch, first draw these points on the coordinate plane. Because the x^2 term has a negative in front of it, the graph will open down like a frown (yes it rhymes), so connect the points (6,0) and (-4,0) with a downward facing parabola.
That's it, you're done!
It factorises to g(x) = (4 + x)(6 - x)
4*6 = 24 and 6 - 4 = 2
so the x intercepts are at -4 and 6
the vertex is half way between the x intercepts wheree x = 1 (so y = 25)
and the y intercept is the constant = 24
(oh and its negative so like a frown (concave down)
The simplest way of all is to make an x y table. Then plot according to the values on the table.
Example: when x=0, then y=24. Then, plot on the graph (0,24) as one of the points.
This is, by the way, how you sketch any graph.
rewriting as g(x) = -x^2 + 2x + 24
g(x) = (-x + 6)(x + 4)
or, g(x) = -(x^2 - 2x + 24) = -(x - 6)(x + 4)
the x-intercepts will be at x = 6 and x = -4
the line of symmetry will be half-way between these at x = 1
when x = 1, y = 25==> the vertex will be at (1,25)
you have the starting points:
vertex at (1,25)
zeros at (6,0) and (-4,0)
y-intercept at (0,24), by symmetry, (2,24) will be on curve
this will be a parabola that opens downward because of the -x^2
you can find some other points to plot by letting x = 3 (and plot this y-value at x = -1 by symmetry)
factor 24 + 2x - x² as (6 - x)(4 + x), so you can plot x = 6 and x = -4 as x intercepts.
line of symmetry is halfway between at (6-4)/2 = x = 1.
vertex on line of symmetry is y = (6-1)(4+1) = 25, so at (1,25). graph opens down.
also y intercept is (0,24), and by symmetry (2,24) is also on the graph.