Finding the Turning Point and Y-Intercept of a Parabola and Linear Graph

When dealing with quadratic equations, understanding the turning point (vertex) of a parabola and the y-intercept of a linear graph is fundamental. This article will explore how to determine these values for the given functions y x^2 - 2x - 3 and y x1.

Turning Point of the Parabola

The turning point of a parabola is its vertex. For the quadratic equation y x^2 - 2x - 3, we can find the vertex (turning point) by completing the square or using calculus.

Method 1: Completing the Square

To complete the square, we rewrite the equation in the vertex form y a(x - h)^2 k, where (h, k) is the vertex.

Rewriting y x^2 - 2x - 3 in vertex form:

Take the quadratic and linear terms: x^2 - 2x.

To complete the square, take the coefficient of x, which is -2, divide by 2, and square it. (-2/2)^2 1.

Complete the square: x^2 - 2x 1 - 1 - 3. This can be rewritten as (x - 1)^2 - 4.

Therefore, the equation in vertex form is: y (x - 1)^2 - 4.

The vertex is at (1, -4), so the x-coordinate of the turning point is 1 and the y-coordinate is -4.

Using Calculus:

First, differentiate the function with respect to x: y' 2x - 2.

Set the derivative to zero to find the critical points: 2x - 2 0. Solving for x, we get x 1.

Substitute x 1 back into the original equation to find the corresponding y-coordinate: y (1)^2 - 2(1) - 3 1 - 2 - 3 -4.

Thus, the turning point is at (1, -4).

Y-Intercept of the Linear Graph

For the linear equation y x1, the y-intercept is the point where the line crosses the y-axis (where x 0). In the form y mx b, b is the y-intercept. Here, the equation is already in slope-intercept form.

Slope and y-intercept analysis:

The equation y x1 is equivalent to y x 1. Here, m 1 and b 1.

The y-intercept is simply the value of y when x 0, which is 1.

This can be easily seen from the graph of the line.

Graphical Representation:

A plot of the graphs for y x^2 - 2x - 3 and y x 1 looks something like this:

Note: A graphical representation would be included here, but since this is a text-based format, we'll refer to it as a hypothetical graph illustrating the parabola and the line intersecting at different points.

Conclusion

By completing the square or using calculus, we can determine the turning point (vertex) of the parabola y x^2 - 2x - 3, which is at (1, -4). For the linear graph y x 1, the y-intercept is simply 1, as it is the constant term in the equation.

Keywords: Parabola turning point, y-intercept, vertex form