Forms of a quadratic function

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Chapter 5.1

Explain the advantage of each of the three forms of a quadratic function (general, factored, and standard form (also known as vertex form)), when graphing.

 

Rewrite the quadratic function in standard (vertex) form,
and give the vertex.

 

Determine whether has a minimum or maximum value and find the value of the minimum or maximum. Also, find the axis of symmetry.

 

Determine the domain and range of . Enter the solution in interval notation.

 

Find the general form of the quadratic function that has a vertex of (-2,-1) and a point on the graph (-4,-3).

 

 

A student converts from standard form to general form. What mistake do you think they made?

 

 

 

Graph the following. Find the vertex, the axis of symmetry, the x and y-intercepts. Give the domain and range, and find where the graph is increasing and decreasing.

 

 

 

 

 

 

 

 

 

 

 

 

 

Find the dimensions of a rectangular corral split into 2 pens of the same size producing the greatest possible enclosed area given 300 feet of fencing.

 

A ball is thrown in the air from the top of a building. Its height, in feet above ground, as a function of time, in seconds, is given by .

Find the vertex (h, k).

Explain the meaning of the vertex in the context of this problem.

 

How long until the ball hits the ground?
a) Write an equation for the graphed function in general form.

 

 

 

 

 

 

 

 

 

b) Using the graph write the equation in standard form.

 

 

In the graph from Q11 above a student writes the following:

“graph increases over the interval (-∞, 8) and decreases over the interval (8, ∞)”
Is this correct? If not, explain what the student has done wrong, and write the correct answer.

 

Find c such that has a maximum value of 15.

Find b such that has a maximum value of 8.

 

5.2

Find the degree and leading coefficient for:

 

Find the x and y-intercepts of the following

 

 

What is the least possible degree of the graphs below? Is the leading coefficient positive or negative?

 

Determine the x-intercepts and end behavior of

 

 

 

What is the relationship between the degree of a polynomial function and the maximum number of turning points in its graph?

How is the range of a polynomial function related to the degree of the polynomial?

A student tries to find the degree of two polynomial functions, one in general form and one in factored form. The two polynomials are as follows

a)
b)

The student describes f(x) as degree 6 and g(x) as degree 5. Explain what mistakes the student made and provide the correct answers.

 

What can we conclude if the graph of a polynomial function exhibits the following end behavior?
a) ,

b) ,

A student graphs the following function

 

 

 

 

 

 

 

Another student says this is all wrong. Can you explain what is correct or what is incorrect about this graph in terms of intercepts, degree, leading coefficient, end behavior etc. What would be the correct function to match the graph?

 

 

 

5.3

For the following functions, state the degree, find the zeros of the polynomial, and state the multiplicity of each.

 

 

Graph the functions. Show all calculations and intercepts.

 

 

 

 

 

 

 

 

 

Find a polynomial function of least degree.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Is it possible for the graph of a polynomial function to have no y-intercept? No x-intercepts? Explain your answer.

 

 

Find a polynomial function of least degree. Leave in factored form.
Degree 4, root of multiplicity 2 at x = 4, root of multiplicity 1 at x = -1, root of multiplicity 1 at x = -3, y-intercept of (0, 144)

 

 

 

 

5.4 and 5.5

Use the Factor Theorem to find all the zeros of given that (x – 3) is a factor. (Synthetic division). Show your work!

 

 

 

Is it possible for a third-degree polynomial with rational coefficients to have no real zeros? Why or why not?

 

Polynomial Inequalities

Solve and write in interval notation showing your work.

 

 

Find a polynomial inequality with the solution of: [-4,3]∪[7,∞)

 

 

 

5.6

For the graphs of the given functions; Show all your work
Find the horizontal and vertical asymptote(s) of the graphs of the given functions.
Find the x and y-intercepts of the rational functions

 

 

 

 

Graph the function, showing all asymptotes. List the x- and y-intercepts. Show your work.

Can a graph of a rational function have no x-intercepts? No y-intercepts? If so, how?

Find a rational function that satisfies the given conditions. Answers may vary, but try to give the simplest answer possible.

Vertical asymptotes x = -3, x = 4; horizontal asymptote y = 2; intercepts (2, 0), (-2, 0) and (0, 2/3)

 

Vertical asymptote at x = 1 and x = 9; intercepts at (7, 0), (3, 0) and (0, 42)

 

 

Find the rational function

 

 

 

 

 

 

 

From the graph above, as

 

 

Sample Solution

came to power in 1979 and represented for many, laissez-faire economics and individual self-determination (Steele, 2018). She believed in power of the market, utilizing it to restore the stagnant British economy and moving away from state provided services. In 1979, cuts resulted in reducing the standard rate of tax from 33% to 30%, the top rate from 83% to 60% and finally cutting public spending by 3% (Bolick, 1995). She reduced the amount of public spending, from 50% to 43%. Thatcher felt high taxes discouraged the incentive to work however, effects of tax cuts increased income inequality through as high earners saw ‘the top 10%- did far better, with their incomes increasing from the equivalent of £472.98 in 1979 to £694.83 in 1990’. The uneven distribution of wealth saw the poorest families receive the least. Reductions in public expenditure affected health, education and social services which created a knock-on effect with substantial loss of public sector jobs resulting in decreased spending on goods and services. Privatisation became Thatcher’s most important and long-lasting legacy. She revealed in her memoirs that it was crucial for ‘reversing the corrosive and corrupting effects of socialism’ Parker. In the 1980-90s, due to fiscal pressures, Thatcher’s conservative views on private ownership and public discontent with the current regime saw the privatisation of public owned entities. For example, the sale of just ‘over 50% of shares in BT and the sale of British Energy in 1996’ (Berrington, 1998). Other privatised industries included electricity, gas, British steel, public bus transportation and other public services. As a result, workforces declined as ‘employment in the electricity and gas industries was cut in half’(Edwards, 2017), problems arose in the regulation of private monopolies to prevent abuse of power, however improved ‘economic growth and improved living standards as privatised businesses cut costs, increased service quality’ (Edwards, 2017). Thatcher can be seen as the key instigator of the sweeping shift from traditional to ‘New Public Management’ initiated by public service reforms. NPM involved the adoption of private sector management ideas to improve structures and processes in the public sector. Thatcher who led the 1980s ‘New Right’ administrations, that put a ‘shrinking government and reduced taxation on the agenda’ (Ferlie, 2017). Thatcher also wanted to remove ‘inefficiency in the state bureaucracy and the deprivilege of the civil service’ as she concluded that the public sector was ‘wasteful, overbureaucratic and underperforming’ (Ferlie et al., 1996). Thatcher wanted to identify areas of waste and inefficiency in the government and ‘improve service quality and customer-orientated service’ (Pollitt, 1996) whilst reducin

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