Question

\(h(x)=-5 x^4+7 x^3-6 x^2+9 x+2\)

Short answer

The polynomial \(h(x)=-5 x^4+7 x^3-6 x^2+9 x+2\) is a quartic function with a leading term of \( -5x^4 \). Its graph falls to negative infinity at both ends due to its degree and leading term. It crosses the y-axis at 2. The roots can be determined with a graphing tool but are non-rational numbers and cannot easily be expressed.

Step-by-step solution

Understand the Task

The problem does not specify a certain task. However, the best course of action is to graph the function and analyze its basic properties, like degree, leading term, end behavior, and intercepts. The roots could be identified using graphing tools, since solving for roots in this case would be very complex.

Identify the Degree and Leading Term

The degree of the function is the highest power in the polynomial, here 4. The leading term in this polynomial is \( -5x^4 \), which determines the end behavior of the graph. Since the exponent is even and the leading coefficient is negative, the graph will go down to negative infinity at both ends.

Identify the y-intercept

The y-intercept is the constant term in the polynomial. So the y-intercept is 2. You find this intercept by setting x=0, which reduces the function to h(0)=2.

Graph and Find Roots

Graphing the function could be performed with a graphing tool. The roots (x-intercepts) could be found where the graph crosses the x-axis. However, these roots are not rational numbers, so identifying them exactly would be difficult without numerical techniques. For this level, it is enough to know that the roots exist and can be found with more advanced methods.

Key concepts

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables raised to whole number powers and their coefficients, summed up together. For instance, the function in question is defined as:
\( h(x) = -5 x^4 + 7 x^3 - 6 x^2 + 9 x + 2 \).
It encompasses terms with varying powers of the variable 'x', each of which contributes to the shape of the graph. Polynomials can model a wide range of phenomena, so learning to graph them is a vital skill in mathematics. This particular polynomial has a quartic term (a term with a power of four), which is the dominant term as 'x' becomes very large in absolute value, thus influencing most of the graph's overall shape.

Degree of a Polynomial

The degree of a polynomial function is an important feature that impacts the function's graph significantly. It is signified by the highest power of 'x' in the function. In \( h(x) = -5 x^4 + 7 x^3 - 6 x^2 + 9 x + 2 \), the degree is 4 because the highest power of 'x' is 4. The degree tells us several things: for instance, it gives us a minimal estimate on the number of times a polynomial's graph will turn, known as turning points, and it's indicative of the possible number of real roots the polynomial can have. Typically, a polynomial can have as many roots as its degree, though some may be complex (non-real) numbers.

End Behavior of a Graph

The end behavior of a graph describes the direction in which the graph heads as 'x' moves towards positive or negative infinity. In the function \( h(x) = -5 x^4 + 7 x^3 - 6 x^2 + 9 x + 2 \), the leading term is \( -5x^4 \). Since the highest power is even and the leading coefficient is negative, the graph will descend towards negative infinity on both sides as 'x' moves away from zero. This knowledge helps in sketching a rough picture of the graph's trajectory before even plotting specific points.

Finding Roots of Polynomials

The roots of a polynomial, also called zeroes or x-intercepts, are the values of 'x' where the polynomial equals zero. They are the points where the graph of the polynomial crosses the x-axis. For a polynomial with a degree as high as 4, finding roots analytically can be quite challenging. One can use numerical methods or graphing calculators to approximate these roots. For example, the polynomial \( h(x) \) provided in the exercise has roots that are not easily identifiable because they are not rational numbers. This is common with higher-degree polynomials.

Y-intercept

The y-intercept of a graph is the point where the graph crosses the y-axis. This occurs when 'x' is set to zero. The y-intercept of a polynomial function is simply the constant term of the function, since all other terms vanish when 'x' equals zero. For the given polynomial \( h(x) \), by setting 'x' to zero, we find the y-intercept to be \( h(0) = 2 \). This is a useful starting point for graphing the polynomial and is easily located on the graph at point (0, 2).