UNIT 3 - Winston-Salem/Forsyth County Schools



ALGEBRA II: UNIT 3

Example 1: The graph of y = x2 – 2x – 3 is shown on the axes below.

[pic]

(a) Draw the graph of y = 5 on the same axes.

(b) Use your graph to find:

(i) the values of x when x2 – 2x – 3 = 5

(ii) the value of x that gives the minimum value of x2 – 2x – 5

Example 2: The diagram below shows part of the graph of y = ax2 + 4x – 3. The line x = 2 is the axis of symmetry. M and N are points on the curve, as shown.

[pic]

(a) Find the value of a.

(b) Find the coordinates of

(i) M;

(ii) N.

Example 3: The graph of the function

y = x2 – x – 2 is drawn below.

[pic]

(a) Write down the coordinates of the point C.

(b) Calculate the coordinates of the points A and B.

Example 4:

(a) Factorize 3x2 + 13x −10.

(b) Solve the equation 3x2 + 13x − 10 = 0.

Consider a function f (x) = 3x2 + 13x −10.

(c) Find the equation of the axis of symmetry on the graph of this function.

(d) Calculate the minimum value of this function.

Example 5: The profit (P) in Swiss Francs made by three students selling homemade lemonade is modeled by the function [pic] where x is the number of glasses of lemonade sold.

HINT: USE THE CALCULATOR TO GRAPH [Y=] and Use [2nd] [TRACE] Commands

(a) Copy and complete the table below Check Table After Graphing for Values

|x |0 |10 |20 |30 |40 |50 |60 |70 |

|A |–13 |p |27 |35 |q |r |11 |s |

i) Calculate the values of p, q, r and s.

ii) Graph the function A.

(c) Answer the following, using the graph or equation for Area.

(i) Write down the equation of the axis of symmetry of the curve,

(ii) Find one value of x for a rectangle whose area is 27 m2.

(iii) Using this value of x, write down the dimensions of the rectangle.

(d) (i) On the same graph, draw the line with equation A = 5x + 30.

(ii) Hence or otherwise, solve the equation 4x2 + x – 5 = 0.

Example 7: Consider the graphs of the following functions.

(i) y = 7x + x2;

(ii) y = (x – 2)(x + 3);

(iii) y = 3x2 – 2x + 5;

(iv) y = 5 – 3x – 2x2.

Which of these graphs (i – iv)

(a) has a y-intercept below the x-axis?

(b) passes through the origin?

(c) does not cross the x-axis?

(d) could be represented by the following diagram?

[pic]

Example 8: Let f (x) = 3(x + 1)2 – 12.

(a) Show that f (x) = 3x2 + 6x – 9.

(b) For the graph of f

(i) write down the coordinates of the vertex;

(ii) write down the equation of the axis of symmetry;

(iii) write down the y-intercept;

(iv) find both x-intercepts.

(c) Hence sketch the graph of f.

Example 9: Part of the graph of the function y = d (x −m)2 + p is given in the diagram below.

The x-intercepts are (1, 0) and (5, 0). The vertex, V, is (m, 2).

[pic]

(a) Write down the value of

(i) m;

(ii) p.

(b) Find d.

Example 10:

(a) Express y = 2x2 – 12x + 23 in the form y = 2(x – c)2 + d.

The graph of y = x2 is transformed into the graph of y = 2x2 – 12x + 23 by the transformations

a vertical stretch with scale factor k followed by

a horizontal translation of p units followed by

a vertical translation of q units.

(b) Write down the value of

(i) k;

(ii) p;

(iii) q.

Example 11: Let f (x) = a (x − 4)2 + 8.

(a) Write down the coordinates of the vertex of the curve of f.

(b) Given that f (7) = −10, find the value of a.

(c) Hence find the y-intercept of the curve of f.

Example 12: The quadratic function f is defined by f (x) = 3x2 – 12x + 11.

(a) Write f in the form f (x) = 3(x–h)2 – k.

(b) The graph of f is translated 3 units in the positive x-direction and 5 units in the positive y-direction. Find the function g for the translated graph, giving your answer in the form g (x) = 3(x – p)2 + q.

Example 13: The diagram shows the graph of y = x2 – 2x – 8. The graph crosses the x-axis at the point A, and has a vertex at B.

[pic]

(a) Factorize x2 – 2x – 8.

(b) Write down the coordinates of each of these points

(i) A;

(ii) B.

Example 14:

(a) Solve the equation x2 – 5x + 6 = 0.

(b) Find the coordinates of the points where the graph of y = x2 – 5x + 6 intersects the x-axis.

Example 15:

(a) Find the solution of the equation x2 – 5x – 24 = 0.

(b) The equation ax2 – 9x – 30 = 0 has solution x = 5 and x = –2. Find the value of a.

Example 16:

The figure below shows part of the graph of a quadratic function y = ax2 + 4x + c.

[pic]

(a) Write down the value of c.

(b) Find the value of a.

(c) Write the quadratic function in its factorized form.

Example 17: The graph of the function f : x [pic] 30x – 5x2 is given in the diagram below.

[pic]

(a) Factorize fully 30x – 5x2.

(b) Find the coordinates of the point A.

(c) Write down the equation of the axis of symmetry.

Example 18: The following diagram shows part of the graph of a quadratic function, with equation in the form y = (x − p)(x − q), where p, q[pic] [pic].

[pic]

(a) Write down

(i) the value of p and of q;

(ii) the equation of the axis of symmetry of the curve.

(b) Find the equation of the function in the form y = (x − h)2 + k, where h, k[pic] [pic].

Example 19: The perimeter of this rectangular field is 220 m. One side is x m as shown.

[pic]

(a) Express the width (W) in terms of x.

(b) Write an expression, in terms of x only, for the area of the field.

(c) If the length (x) is 70 m, find the area.

Example 20: A farmer wishes to enclose a rectangular field using an existing fence for one of the four sides.

[pic]

(a) Write an expression in terms of x and y that shows the total length of the new fence.

(b) The farmer has enough materials for 2500 meters of new fence. Show that y = 2500 – 2x

(c) A(x) represents the area of the field in terms of x.

(i) Show that A(x) = 2500x – 2x2

(ii) find the value of x that produces the maximum area of the field.

(iii) Find the maximum area of the field.

Example 21: The diagram shows a path x m wide around a rectangular lawn that measures 10 m by 8 m.

[pic]

(a) Write down an expression in terms of x for the area of the path.

(b) What is the width of the path when its area is 208 m2

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