Quadratic Functions

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What is the general form of a quadratic function?

The general form of a quadratic function is f(x) = ax^2 + bx + c, where a, b, and c are constants.

What is the shape of the graph of a quadratic function?

The graph of a quadratic function is a parabola.

What determines the direction in which a parabola opens?

The sign of the leading coefficient 'a'. If 'a' is positive, the parabola opens upwards; if 'a' is negative, it opens downwards.

What is the axis of symmetry in a quadratic function?

The axis of symmetry is a vertical line that divides the parabola into two mirror-image halves. Its equation is x = -b/(2a).

Where is the vertex of a parabola located?

The vertex is located at the point (h, k). For a quadratic function in standard form, it is at (-b/(2a), f(-b/(2a))).

What are the roots of a quadratic function?

The roots are the x-values where the graph intersects the x-axis. They are also called solutions or zeros of the function.

How can the roots of a quadratic function be calculated?

Using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a.

What is the discriminant in a quadratic function?

The discriminant is the part of the quadratic formula under the square root: b^2 - 4ac. It determines the number of real roots.

What does it mean if the discriminant is positive?

If the discriminant is positive, the quadratic function has two distinct real roots.

What if the discriminant is zero?

If the discriminant is zero, there is exactly one real root, meaning the parabola touches the x-axis at one point.

And if the discriminant is negative?

If the discriminant is negative, there are no real roots, implying the parabola does not intersect the x-axis.

Can quadratic functions be solved by factoring?

Yes, if the quadratic can be factored into a product of binomials, it can be solved using the zero-product property.

What method can be used besides factoring and the quadratic formula to solve quadratic functions?

Completing the square is another method used for solving quadratic functions.

What is the vertex form of a quadratic function?

The vertex form is f(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola.

How does the quadratic function relate to real-world applications?

Quadratic functions model many real-world situations, such as projectile motion, area problems, and maximum/minimum profit calculations.