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The opening direction of a parabola is a fundamental feature to understand when graphing quadratic functions. It tells us if the parabola has a 'smiley face' (opens upward) or a 'frowny face' (opens downward). This characteristic is determined by the sign of the leading coefficient in the quadratic equation of the form f(x) = ax^2 + bx + c. If a is positive, the parabola opens upward, whereas if a is negative, it opens downward.

In our example, with the equation f(x) = x^2 + 4x - 5, the leading coefficient is 1, which is indeed positive. This indicates that our graph will be a 'smiley face' curving upwards from the vertex. Understanding the direction in which a parabola opens is crucial as it affects the position of the vertex and the shape of the graph.

The vertex is the highest or lowest point on a parabola, acting as a pivotal element in its graph. It's the point where the parabola changes direction. To find the vertex of a parabola given by the equation f(x) = ax^2 + bx + c, we use the formula h = -b/(2a) to calculate the x-coordinate and then substitute h back into the function to find the y-coordinate. The vertex form of a parabola is then (h, k) where k = f(h).

In our quadratic function f(x) = x^2 + 4x - 5, by applying the vertex formula, we calculated h = -4/(2*1) = -2 and then found k = f(-2) = -1. So, the vertex of this parabola is (-2,-1). Locating the vertex is an essential step towards sketching the graph since it serves as a reference point.

X-intercepts are the points where the graph of a quadratic function crosses the x-axis, also known as the solutions to the equation. To find the x-intercepts, you set the quadratic equation f(x) = 0 and solve for x. The quadratic formula, x = (-b ± √(b^2-4ac))/(2a), can be used when the equation does not factor easily.

Our function, f(x) = x^2 + 4x - 5, yields the x-intercepts by setting it to zero and solving for x, giving us the points x = -5 and x = 1. These are the points where the parabola meets the x-axis, and they are instrumental in shaping the graph.

The y-intercept is the point where the graph intersects the y-axis, which occurs when x is zero. Simply substitute x with 0 in the quadratic equation to find the y-intercept. It's easy to locate and helps to plot the quadratic graph accurately.

For example, the y-intercept of f(x) = x^2 + 4x - 5 is found by calculating f(0) = 0^2 + 4(0) - 5 = -5. Therefore, the point where our parabola crosses the y-axis is (0,-5). This single value can guide students when they start plotting the parabola on a coordinate plane.