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The vertex of a parabola is a key point that helps to understand the shape and the location of the parabola on a graph. In the quadratic function format \(y=ax^2+bx+c\), the vertex's x-coordinate can be found using the formula \(h=-\frac{b}{2a}\). The vertex is always a turning point in the parabola, where the graph either reaches a maximum or a minimum value, depending on the sign of the coefficient \(a\).
For the given function \(y=5x^2+4x-5\), we calculate the x-coordinate of the vertex by substituting \(a=5\) and \(b=4\) into the vertex formula, yielding \(h=-\frac{4}{10}=-0.4\). Once you have \(h\), plug it into the original equation to find the y-coordinate, which gives \(-5.8\). Therefore, the vertex is \((-0.4, -5.8)\).
This point is crucial because it indicates the lowest point on the graph for this upward-opening parabola.

• The formula for the x-coordinate of the vertex helps locate the parabola efficiently.
• A positive coefficient \(a\) means the parabola opens upwards, indicating the vertex will be a minimum point.

The x-intercepts of a quadratic function are the points where the graph crosses the x-axis. To find these points, we set the function equal to zero: \(0=ax^2+bx+c\).
For our specific equation \(0=5x^2+4x-5\), the x-intercepts are the solutions to this equation. There are several methods to solve this, including factoring, completing the square, or using the quadratic formula. Using the quadratic formula \(x=\frac{-b \pm \sqrt{b^2-4ac} }{2a}\), where \(a=5\), \(b=4\), and \(c=-5\), we find the x-intercepts.
Substituting into the formula gives the roots \(x=-1\) and \(x=1\), which are the points \((-1, 0)\) and \((1, 0)\) on the graph.

• X-intercepts are found by solving the equation \(ax^2+bx+c=0\).
• The solutions provide critical points where the graph touches or crosses the x-axis.
• These intercepts are real and distinct because the parabola crosses the axis at two points.

The quadratic formula is a powerful tool for finding the roots (or x-intercepts) of any quadratic equation of the form \(ax^2+bx+c=0\). It is expressed as \(x=\frac{-b \pm \sqrt{b^2-4ac} }{2a}\). This formula allows for the determination of the x-values where the quadratic function equals zero, which corresponds to where the graph intersects the x-axis.
Using the quadratic formula requires identifying coefficients \(a\), \(b\), and \(c\) from the equation. It is especially useful when the equation does not factor easily.

• If \(b^2-4ac\) (the discriminant) is positive, there are two distinct real roots, meaning the parabola crosses the x-axis twice.
• If the discriminant is zero, there is exactly one real root, and the parabola touches the x-axis at a single point.
• If the discriminant is negative, the roots are complex, and the parabola does not cross the x-axis at all.
• The quadratic formula provides a consistent method to solve any quadratic equation for its roots.

The equation \(5x^2+4x-5=0\) fits this requirement, and by using \(a=5\), \(b=4\), \(c=-5\), we find clear x-intercepts at \(x=-1\) and \(x=1\), corroborating with other methods despite the potential complexities of the equation.