What is a non-linear equation?

A non-linear equation is an equation in which the highest exponent of the variable(s) is greater than one or the variable(s) are multiplied together. Non-linear equations create graphs that are not straight lines.

How do you identify a non-linear function from an equation?

To identify a non-linear function from an equation, look for terms that are squared, cubed, or involve products of variables. For example, y = x² + 3x + 2 is non-linear due to the x² term.

Solve the non-linear equation: x² - 4x + 3 = 0.

Hint: Factor the quadratic if possible.

To solve x² - 4x + 3 = 0, factor it as (x - 1)(x - 3) = 0. Therefore, the solutions are x = 1 and x = 3.

What is the general form of a quadratic function?

The general form of a quadratic function is f(x) = ax² + bx + c, where a, b, and c are constants and a ≠ 0. The graph of a quadratic function is a parabola.

How can you determine the vertex of the quadratic function y = 2x² - 8x + 5?

To find the vertex of the quadratic function y = 2x² - 8x + 5, use the vertex formula x = -b/(2a). Here, a = 2 and b = -8. Thus, x = 8/(2*2) = 2. Plug x = 2 back into the function: y = 2(2)² - 8(2) + 5 = 2(4) - 16 + 5 = 8 - 16 + 5 = -3. The vertex is at (2, -3).

What is the quadratic formula used to find the roots of ax² + bx + c = 0?

The quadratic formula is x = (-b ± √(b² - 4ac)) / (2a). It is used to find the roots of the quadratic equation ax² + bx + c = 0.

What transformation occurs when the equation of a parabola y = a(x - h)² + k is given?

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If the discriminant (b² - 4ac) of a quadratic equation is negative, what can be concluded about the roots?

Hint: Consider the nature of the roots.

If the discriminant (b² - 4ac) is negative, the quadratic equation has no real roots. The roots are complex or imaginary.

What is a rational function and how does it differ from a polynomial function?

A rational function is a function that can be expressed as the ratio of two polynomials, f(x) = P(x)/Q(x), where Q(x) ≠ 0. In contrast, a polynomial function only includes non-negative integer exponents and does not have a denominator that is a polynomial.

What is the significance of the vertex in a quadratic function, and how can it be found?

Hint: Use the formula for the x-coordinate of the vertex.

The vertex of a quadratic function represents the maximum or minimum point of the parabola. It can be found using the formula x = -b/(2a), where a and b are coefficients from the quadratic equation in standard form.