Graphing.. I don't undertand the text.. Please help...?

Question

Complete the given table of values. Identify the x-intercepts, vertex, and the axis of symmetry.
(I don't need an explanation of the graph. I know how to do that. it's the rest of it)

1. y = x^2 + 2x - 3
x y
-2 ?
-1 ?
0 ?
1 ?
2 ?

2. y = x^2 + 1
x y
-2 ?
-1 ?
0 ?
1 ?
2 ?

Answer 1

Hi,

Just plug in each x value and work out the corresponding y values.

1. y = x^2 + 2x - 3
x y
-2 -3
-1 -4 <== this is the vertex, your lowest point with the same
0 -3..........y values on each side of it.
1 0 <== this is an x intercept because its y value is 0.
2 5

The axis of symmetry is found from the formula x = -b/(2a) for the equation y = ax² + bx + c. For this problem, a = 1 and b = 2, so the axis is x = -2/(2*1) = -1. Since the vertex is always on the axis of symmetry, this confirms that the point (-1,-4) is the vertex.

In addition to the x intercept noted above at (1,0), there's another x intercept at x = -3 at the point (-3,0). That's because this graph is symmetrical across the axis of symmetry, so if there's a zero at x = 1, 2 squares to the right of the axis, then there must be another zero 2 squares to the left of the axis at x = -3.

2. y = x^2 + 1
x y
-2 5
-1 2
0 1 < vertex is your lowest point/ same y values each side
1 2
2 5

(0,1) is also your y intercept. There are no x intercepts because the graph's lowest point is (0,1) so it is never low enough to touch the x axis.

The axis of symmetry is found from the formula x = -b/(2a) for the equation y = ax² + bx + c. For this problem, a = 1 and b = 0, so the axis is x = -0/(2*1) = 0. Since the vertex is always on the axis of symmetry, this confirms that the point (0,1) is the vertex.

I hope this helps!! :-)

Answer 2

There arises an assumption that your inquiry is an assigned question to you by your mathematics class... Nevertheless, I am in a state of boredom...

Recall:
f(x) = ax^2 + bx + c is the general form of a quadratic equation

The identification of the X-intercepts (where values of Y are zero) can be obtained through polynomial factorization using the quadratic formula:

x = [-b +- (b^2-4ac)^1/2]/2a

Plug-in the values into the equation to obtain two values for X since:

x = (-2 +- 4)/4

It follows that:

x = (-2+4)/4 and x = (-2-4)/4

Therefore, your two x-intercepts are 1/2 and -3/2.

For the coordinates of the vertex and the whereabouts of the axis of symmetry, we transform the general equation into a parabolic equation.

Given y = x^2 + 2x - 3,
We use the Vertex Form of the quadratic equation f(x) = a(x - h)^2 + k

y = x^2 + 2x - 3

Using, perfect square trinomial: (Recall that (a+b)^2 = (a^2 + 2ab +b^2)

y (+1) = x^2 + 2x (+1) -3 (add 1 to both sides)
(because x^2 + 2x + 1 is the same as x^2 + 2(x)(1) + 1^2)

which gives us the simplified equation:

y = (x + 1)^2 - 4
But where is (a) to complete the Vertex Form?...

Here it is:

y = (1)(x +1)^2 - 4

Now we are able to derive the vertex (h, k) and axis of symmetry (x = h) from the above equation.

Vertex: (x = -1, y = -4) or (-1, -4)
Axis of Symmetry: (Since the variable x is squared, the axis of symmetry is found the x-axis, which is x = -1)

This is only for number one, I am rather confident with your present abilities to solve for number 2.

Kudos ^_^

Answer 3

Ok, let's see

1. x^2 + 2x - 3 = y
(x+3)(x-1) = y
Intercepts of the equation...values for x, y are zero
y is zero for x = -3, 1
x is zero y = -3 (Vertex)
y is the axis of symmetry
WIth the above info you can construct the rest of the chart.

2. y = x^2 + 1
x = 0, y =1
y = 0 only have imaginary solution, so there are no x = 0 intercepts. this is the vertex

Good luck.

Answer 4

To find a value of y, substitute the value of x in the formula. And to find the x-intercept you have to qualify the formula with zero and solve for x (in #1 x^2+2x-3=0 and solve for x). To find the axis of symmetry you have to solve it by completing the square. I'll do #1 for you.
y=x^2+2x-3 (add the sq are of the half of the x coefficient and subtract it and the x^2 coefficient must be 1)
y=x^2+2x+1-1-3
y=(x+1)^2-4
y+4=(x+1)^2
so the axis of symmetry will be x=-1 (the number which inside the square but with an opposite sign)

Answer 5

y = -3, -4, -3, 0, 5

y = 5, 2, 1, 2, 5