How do you find the important points to graph #y= x^2-x-3#?

Question

Avery Alexander · Accepted Answer

#color(blue)("You asked "underline( "'How do you'"))#. So I have shown you the 'tricks' of how to!
I have found the #x_("vertex")# value as demonstration. See if you can make it all work!

#color(green)("Method tutorial")#

It is a quadratic curve (Horse shoe shape)

#color(blue)("General shape")#
The coefficient of #x^2# is +1 #-> positive#
so the shape is U. If the coefficient had been negative it would have been the other way up ( #nn#)
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#color(blue)("The y-intercept")# is when #x=0# so you can find it by substitution. (Tip: it is the constant!!!!)
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#color(blue)("The x-intercept")# is when #y=0# so you can find this by factoring (much the quicker method) or using the formula:

Standard form equation of #ax^2+bx+c=0#

In conjunction with # x= (-b+-sqrt(b^2-4ac))/(2a)#

If the #sqrt(b^2-4ac)# bit is negative then the curve does not cross the x-axis and if you are determined to get some answers you have to use a branch of maths called Complex Numbers.

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#color(blue)("Turning point of the curve (vertex)")#

The Top/Bottom of the curve (maximum/minimum) is quite easy to find. Using the standard form equation. Change it to

#y=a(x^2+b/ax) +c#

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#color(brown)("There another one you can use called vertex equation which has a"#
#color(brown)("different approach")#
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In your case a=1 so #b/a x ->color(blue)((-1)/1) x #

#color(green)(v_("vertex"))=(-1/2)xxcolor(blue)(b/a) -> (-1/2)xx(color(blue)(-1)) =color(green)(+1/2)#

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#color(green)(y_("vertex")) ->#substitute the found value of #color(green)(v_("vertex"))#

Hazel Anglin · Answer

undefined To graph the function y = x^2 - x - 3, you can find important points by:

1. Calculating the vertex of the parabola using the formula x = -b/(2a), where a = coefficient of x^2, and b = coefficient of x.
2. Finding the y-coordinate of the vertex by substituting the x-coordinate into the function.
3. Determining the y-intercept by setting x = 0 and solving for y.
4. Locating the x-intercepts by setting y = 0 and solving the quadratic equation for x.
5. Plotting these points on a coordinate plane and drawing the parabolic curve passing through them.