Graphing Polynomials
Decide whether each of the following is sometimes true, always true, or never true and explain reasoning.
A. A cubic function has 2 different x-intercepts
^Never true, has to have 3?
B. A cubic function has 3 x-intercepts
^ Always true
C. A quartic function has the range of all real numbers
^ ?
D. A quartic function has no x-intercepts
^Never true
E. A quadratic function will have at most 2 x-intercepts
Always true
F. A quintic function and a cubic function has the same range
?
A. sometimes true: x(x-1)^2
B. Sometimes true x^3 ; x(x-1)(x-2)
C. never. It always opens up or down.
D. sometimes true. x^4+1
E. always true
F. true. Always has range of all reals.
Every cubic looks like x^3, and every quintic looks like x^5, with some wiggles around the roots, but in the end they all go from -infinity to infinity.
A. A cubic function has 2 different x-intercepts:
This statement is never true. A cubic function will always have either 3 x-intercepts or 1 x-intercept. It cannot have exactly 2 different x-intercepts because by the Fundamental Theorem of Algebra, a cubic polynomial will have either 3 real roots or 1 real root and a pair of complex conjugate roots.
B. A cubic function has 3 x-intercepts:
This statement is always true. A cubic function is a polynomial of degree 3, and according to the Fundamental Theorem of Algebra, it will have exactly 3 roots or x-intercepts. These x-intercepts can be real or complex, but there will always be exactly 3.
C. A quartic function has the range of all real numbers:
This statement is sometimes true. The range of a quartic function can be all real numbers, but it is not always the case. The range depends on the leading coefficient and the behavior of the function at its extremes. If the leading coefficient is positive, the range will be all real numbers from negative infinity to positive infinity. However, if the leading coefficient is negative, the range will be restricted. For example, if the leading coefficient is negative, the range of the quartic function will be bounded above.
D. A quartic function has no x-intercepts:
This statement is never true. A quartic function, which is a polynomial of degree 4, will always have at least 1 x-intercept. It may have more than 1, but it cannot have zero x-intercepts. This is because a quartic function will always cross the x-axis at least once or touch it.
E. A quadratic function will have at most 2 x-intercepts:
This statement is always true. A quadratic function, which is a polynomial of degree 2, will have at most 2 x-intercepts. It may have 2 distinct real roots, 1 repeated real root, or no real roots at all (in the case of complex roots).
F. A quintic function and a cubic function have the same range:
This statement is not always true. The range of a function depends on its coefficients and behavior, so it is possible for a quintic function and a cubic function to have the same range. However, it is also possible for them to have different ranges. The range of a function can be affected by the leading coefficient, the constants, and the degree of the polynomial. Therefore, without specific information about the functions in question, it is not possible to determine whether they will have the same range or not.
A. A cubic function has 2 different x-intercepts:
This statement is never true. A cubic function is a polynomial of degree 3, which means it can have at most 3 x-intercepts. It is possible for a cubic function to have 2 or 3 x-intercepts, but it cannot have just 2 x-intercepts.
B. A cubic function has 3 x-intercepts:
This statement is always true. A cubic function is a polynomial of degree 3, and according to the Fundamental Theorem of Algebra, a polynomial of degree n will have exactly n x-intercepts. Therefore, a cubic function will always have 3 x-intercepts.
C. A quartic function has the range of all real numbers:
This statement is sometimes true. The range of a quartic function depends on the coefficients and the leading term of the function. If the quartic function has a leading term with a positive coefficient (like x^4), then the range will be all real numbers. However, if the leading term has a negative coefficient (like -x^4), then the range will be restricted and not include all real numbers.
D. A quartic function has no x-intercepts:
This statement is never true. A quartic function is a polynomial of degree 4, which means it can have up to 4 x-intercepts. It is possible for a quartic function to have no x-intercepts, but it is not always the case.
E. A quadratic function will have at most 2 x-intercepts:
This statement is always true. A quadratic function is a polynomial of degree 2, and according to the Fundamental Theorem of Algebra, a polynomial of degree n will have at most n x-intercepts. Therefore, a quadratic function will have at most 2 x-intercepts.
F. A quintic function and a cubic function have the same range:
This statement is uncertain. The range of a function depends on its coefficients and other factors. Without specific information about the quintic and cubic functions in question, it is difficult to determine whether their ranges will be the same.