If f and g have the same sign where they intersect, is f times g zero?
If f and g intersect on the x axis then they both have value zero and for that x value, f * g will be 0 * 0 = 0.
If they intersect anywhere else then they have identical nonzero values and, since the product of any real number with itself is positive, their product will be positive.
What is the equation for doubling time?
A function f(t) doubles in time tDoub, starting from clock time t, if f(t + tDoub) = 2 * f(t). For any function but an exponential function, tDoub will be different for different values of t.
The exponential function f(t) = A b^t doubles in the same time tDoub no matter what the starting t. The equation is still
f(t + tDoub) = 2 * f(t); for an exponential function this equation is
A b^(t + tDoub) = 2 * A b^t. Dividing both sides by a and writing b^(t + tDoub) as b^t * b^tDoub, we get
b^t * b^tDoub = 2 * b^t. Dividing both sides by b^t we have
b^tDoub = 2. If we take the log of both sides we get
tDoub log(b) = log(2) so that
tDoub = log(2) / log(b).
See more at the following links:
11-02-2005_____doubling_time_given_growth_rate
10-25-2005_____doubling_time_at_rate_57_percent
11-04-2005_____solving_doubling_time_equation_for_t____use_trial_and_error_at_this_point
11-08-2005_____doubling_time_starting_at_t0
11-13-2005_____another_explanation_of_doubling_time
When it ask for us to linearize a data set, does that mean for us just to graph a line through it?
To linearize a data set:
Graph log(y) vs. log(x), y vs. log(x) and log(y) vs. x.
If one of the graphs comes out linear, find the slope and vertical intercept and write down the corresponding linear function (e.g, log(y) = m log(x) + b, if the log(y) vs. log(x) graph is the one that comes out linear).
Solve the equation you get for y.
See also the links at the 23-24-242 site:
12-10-2005_____construct_a_table_for_given_function_and_linearize
(11-17-2005_____linear_fits___logarithmic_equations)
12-06-2005_____linearizing_a_data_set.
What are the types of graphs that a polynomial of degree less than 4 can have?
See also the links
12-01-2005_____questions_related_to_polynomials
12-06-2005_____graph_of_a_polynomial_with_distinct_linear_factors
12-03-2005_____zeros_of_a_polynomial___forms_of_a_polynomial
12-06-2005_____how_to_find_zeros_of_linear_polynomials
12-06-2005_____what_do_you_mean_by_a_quadratic_polynomial
12-06-2005_____explaining_the_graph_of_a_polynomial_function
12-06-2005_____quad_polynomial_with_no_zeros_is_not_product_of_two_linear_functions
12-06-2005_____what_quadratic_function_describes_behavior_of_polynomial_near_degree_2_zero
12-10-2005_____what_quadratic_function_describes_behavior_near_this_zero_of_a_polynomial
12-09-2005_____graphing_a_factored_polynomial_with_negative_leading_coefficient
12-09-2005_____graphing_a_specific_factored_polynomial
12-09-2005_____meaning_of_distinct_and_repeated_linear_factors_and_their_effect_on_a_graph
12-10-2005_____sketching_a_specific_polynomial__
12-10-2005_____what_are_irreducible_quadratic_factors___fund_thm_of_algebra___graphing_polynomials
Once you understand how the graph of a polynomial is related to its linear and irreducible quadratic factors, the following summarize the possible shapes of polynomial of degree 4 or less:
A polynomial of degree 0 is a constant function whose graph is a horizontal line at y = c, where c is the constant.
A polynomial of degree 1 is a linear function and forms a straight line which intersects the x axis at one point.
A polynomial of degree 2 is a quadratic function and forms a parabolic graph which might intersect the x axis at 0, 1 or 2 points, depending on the location of its vertex and whether it opens upward or downward.
A polynomial of degree 3 with no irreducible quadratic factors has three linear factors.
If they are all distinct then the graph passes through the x axis in 3 points.
If two linear factors are the same and one is different then at the zero indicated by the common factors it will touch the x axis like a parabola at its vertex, without passing thru the x axis, and will pass pretty much straight through the x axis at the other zero.
If all three linear factors are the same then the graph passes thru the corresponding zero the same way the y = x^3 power function passes thru the origin.
If the polynomial of degree 3 has an irreducible quadratic factor, then it has only one linear factor and only one zero, which is passes straight through.
A polynomial of degree 4 with no irreducible quadratic factors has four linear factors.
If the four linear factors are all distinct then the graph passes through the x axis in 4 points.
If two linear factors are the same and the other two are different from it and from each other, then at the zero indicated by the common factors the graph will touch the x axis like a parabola at its vertex, without passing thru the x axis, and will pass pretty much straight through the x axis at the other two zeros.
If two linear factors are the same and the other two are different from it but are also the same, then at each of the two zeros on the x axis the graph will touch the x axis like a parabola at its vertex, without passing thru the x axis.
If three linear factors are the same and the other is different then the graph passes thru the 'triple' zero the same way the y = x^3 power function passes thru the origin, and passes straight through the other.
If all four linear factors are the same then the graph passes thru the corresponding zero the same way the y = x^4 power function passes thru the origin.
If the polynomial of degree 4 has an irreducible quadratic factor, then it has only two linear factors. If they are distinct then the graph passes pretty much straight through the x axis at each point. If they are the same then the graph touches the x axis at the corresponding zero the same way the y = x^2 parabola touches the x axis without passing through.
How do we find the vertex from the equation y=.028t^2+-1.4t+69?
The vertex is found on the vertical line x = -b / (2a). The y coordinate is found by substituting the x coordinate into the equation.
See also
09-21-2005_____vertex_and_basic_points_of_shifted_or_stretched_basic_parabola
09-21-2005_____does_quadratic_model_work_for_two_data_points_if_we_plug_in_zero_for_the_other
09-21-2005_____finding_the_vertex
09-21-2005_____what_is_the_vertex_and_how_do_we_find_it
Does a stretch happen when there is a number in front of the parenthesis, and is the shift the c of the equation?
In the form y = A f(x-h)+c, A is the vertical stretch and c the vertical shift. h is the horizontal shift.
Examples:
y = 3 ( x - 7 ) ^ 2 + 5 has vertical stretch factor 3, vertical shift 5 and horizontal shift 7.
y = 3 * 2^(t - 5) + 6 has vertical stretch factor 3, vertical shift 6 and horizontal shift 5.
09-21-2005_____write_function_to_stretch_basic_exp_fn_vertically_then_shift_horizontally
09-22-2005_____sketching_an_infinite_family_of_straight_lines_with_constant_slope
09-23-2005______Stretching_then_horiz_shifting_then_vert_shifting_exp_fn____how_to_do_all_three
09-21-2005_____sketching_the_graph_of_an_exponential_family
If a quadratic polynomial f(x)=ax^2+bx+c has no zeros, then why is that polynomial not a product of two linear polynomials?
A linear polynomial has a zero. When you multiply two linear polynomials each one results in a zero. So if the quadratic polynomial was the product of two linear factors, it could not fail to have zeros.