Please, can anyone explain the concept of graphing quadratic functions to me?

For example, how must I graph gradratic functions such as y = x^2 - 2x - 3?

If anyone could also include step-by-step solving methods, I would truly appreciate it.

If you know very little about the topic, a good way would be to simply make a table of values for x and y

pick reasonable small values for x and evaluate to get the y

e.g.

x y
0 -3
1 -4
2 -3 ------> 2^2 - 2(2) - 3 = 4-4-3 = -3
3 0
4 5 ----> 4^2-2(4)-3 = 16-8-3 = 5
5 12
-1 0
-2 5
-3 12 ---> (-3)^2 - 2(-3)-3 = 9+6-3 = 12

now you have 9 points, plot them and you should see the outline of the parabola. Join the points with a smooth curve, don't join them with straight lines.
Notice that (1 , -4) is the vertex of the parabola

your graph should look like this

http://www.wolframalpha.com/input/?i=plot+y+%3D+x%5E2+-+2x+-+3

Reiny's method is fine, but the "standard" method is like this:

Find the vertex, which is the highest or lowest point of the parabola. Use -b/2a first.

y = ax^2 + bx + c

y = x^2 - 2x - 3

2/2 = 1, axis of symmetry.

y = 1^2 - 2(1) - 3
y = 1 - 2 - 3
y = -4

Vertex: (1, -4)

y-intercept: y = 0^2 - 2(0) - 3

y-intercept: -3

Choose another number that is on the same side as the y-intercept and vertex.

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

y = 16 - 8 - 3

y = 5

(4, 5) is another point.

Graph using the points (4,5) and (1,4) and the y-intercept -3.

Sure! Graphing quadratic functions involves plotting points on a coordinate plane to create a curve called a parabola. To graph a quadratic function, such as y = x^2 - 2x - 3, you can follow these step-by-step methods:

Step 1: Set up a coordinate plane. Draw the x and y axes, and label them accordingly.

Step 2: Determine the vertex. The vertex of a quadratic function can be found using the formula x = -b/(2a), where a, b, and c are the coefficients of the quadratic equation (in the form ax^2 + bx + c = 0). In your case, the equation is y = x^2 - 2x - 3, so a = 1, b = -2, and c = -3. Plugging these values into the formula, we find x = -(-2)/(2*1) = 2/2 = 1. To find the y-coordinate of the vertex, plug x = 1 into the equation: y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4. Therefore, the vertex is at (1, -4).

Step 3: Find the y-intercept. To determine the y-intercept, substitute x = 0 into the equation and solve for y: y = (0)^2 - 2(0) - 3 = -3. So, the y-intercept is at (0, -3).

Step 4: Find the x-intercepts (if they exist). The x-intercepts are the points where the graph crosses or touches the x-axis. To find these points, set y = 0 in the equation and solve for x: 0 = x^2 - 2x - 3. This quadratic equation can be factored as (x - 3)(x + 1) = 0. Therefore, the x-intercepts are x = 3 and x = -1.

Step 5: Plot the points. Plot the coordinates for the vertex, y-intercept, and x-intercepts on the coordinate plane. In this case, plot the points (1, -4), (0, -3), (3, 0), and (-1, 0).

Step 6: Draw the parabola. Use the plotted points to sketch a smooth curve that passes through these points. The parabola will open upwards because the coefficient of x^2 is positive. The graph should look like a "U" shape.

Step 7: Optional - Find the axis of symmetry. The axis of symmetry is a vertical line that passes through the vertex of the parabola. In this case, the vertex is at (1, -4), so the axis of symmetry is the vertical line x = 1. You can draw a dashed line to represent the axis of symmetry.

That's it! You have successfully graphed the quadratic function y = x^2 - 2x - 3. Feel free to label the graph, add the equation, or make any additional markings you find helpful.