QUADRATIC FUNCTIONS Investigating Quadratic Functions in Verte Form The two forms of a quadratic function that have been eplored previousl are: Factored form: f ( ) a( r)( s) Standard form: f ( ) a b c In this unit, we will be working with quadratic functions in verte form, where a, p, and q are constants, a. Verte form: f ( ) a( p) q We will continue to use the following vocabular associated with quadratic functions: Parabola: The smmetrical curve of the graph of a quadratic function. Verte of a Parabola: The lowest point on the graph if the graph opens upward or the highest point on the graph if the graph opens downward. Minimum / Maimum Values: If the verte is the lowest point on the graph, then the -coordinate of the verte is called the minimum value. It is the smallest value in the range of the function. If the verte is the highest point on the graph, then the -coordinate of the verte is called the maimum value. It is the greatest value in the range of the function. Ais of Smmetr: A vertical line through the verte that divides the graph of a quadratic function into two congruent halves. The -coordinate of the verte defines the equation of the ais of smmetr. Eample : Investigate the Basic Quadratic Function For the function, complete the table of values and sketch the graph on the grid provided. Compare the equation to the form a( p) q and state the values of a, p, and q. State the verte coordinates, maimum or minimum value, equation of the ais of smmetr, domain and range of the function.. Verte Ma / Min Value Ais of Smmetr Note: When sketching this basic parabola, notice that the points that were plotted can be described in terms of their relative position to the verte. That is, from the verte, points are located: over unit, up unit over units, up units over units, up units
Eample : a. For the function, complete the table of values and sketch the graph on the grid provided. Compare the equation to the form a( p) q and state the values of a, p, and q. State the verte coordinates, maimum or minimum value, equation of the ais of smmetr, domain and range of the function. 9 9 Verte Ma / Min Value Ais of Smmetr for? c. Describe the graph of as a transformation of. Eample : a. For the function, complete the table of values and sketch the graph on the grid provided. Compare the equation to the form a( p) q and state the values of a, p, and q. State the verte coordinates, maimum or minimum value, equation of the ais of smmetr, domain and range of the function. 9 9 Verte Ma / Min Value Ais of Smmetr for? c. Describe the graph of as a transformation of.
Eample : a. For the function, complete the table of values and sketch the graph on the grid provided. Compare the equation to the form a( p) q and state the values of a, p, and q. State the verte coordinates, maimum or minimum value, equation of the ais of smmetr, domain and range of the function. 9 9 for? Verte Ma / Min Value Ais of Smmetr c. Describe the graph of as a transformation of. Eample 5: a. For the function, complete the table of values and sketch the graph on the grid provided. Compare the equation to the form a( p) q and state the values of a, p, and q. State the verte coordinates, maimum or minimum value, equation of the ais of smmetr, domain and range of the function. 9 9 Verte Ma / Min Value Ais of Smmetr for? c. Describe the graph of as a transformation of.
Eample 6: a. For the function, complete the table of values and sketch the graph on the grid provided. Compare the equation to the form a( p) q and state the values of a, p, and q. State the verte coordinates, maimum or minimum value, equation of the ais of smmetr, domain and range of the function. 9 9 Verte Ma / Min Value Ais of Smmetr for? c. Describe the graph of as a transformation of. Eample 7: a. For the function, complete the table of values and sketch the graph on the grid provided. Compare the equation to the form a( p) q and state the values of a, p, and q. State the verte, maimum or minimum value, equation of the ais of smmetr, domain and range of the function. 9 9 9 9 Verte Ma / Min Value Ais of Smmetr for? c. Describe the graph of as a transformation of.
Eample 8: a. For the function, complete the table of values and sketch the graph on the grid provided. Compare the equation to the form a( p) q and state the values of a, p, and q. State the verte, maimum or minimum value, equation of the ais of smmetr, domain and range of the function. 9 9 9 9 Verte Ma / Min Value Ais of Smmetr for? c. Describe the graph of as a transformation of. Conclusions: For a quadratic function written in the form a( p) q graph of the basic quadratic function,, are as follows:, the effects of the parameters a, p, and q on the If a >, then the parabola opens. If a <, then the parabola opens. If < a <, then the parabola is compared to the graph of If a < or a >, then the parabola is compared to the graph of We sa that the graph of If a is negative, then we sa that the graph of has been verticall stretched b a factor of a.. has been reflected through the -ais.. If q >, then the parabola is shifted compared to the graph of If q <, then the parabola is shifted compared to the graph of We sa that the graph of has been verticall translated q units. If p >, then the parabola is shifted compared to the graph of If p <, then the parabola is shifted compared to the graph of We sa that the graph of has been horizontall translated p units.....
Combining Transformations Eample 9: Sketch the graph of the function maimum or minimum value, domain and range. Method : ( ) and state the verte, ais of smmetr, direction of opening, Determine the mapping rule. Substitute points from the table of values for the base function, mapping rule to obtain points for the new function. Sketch the parabola., into the Mapping Rule: 9 9 ( ) Method : Plot the verte on the grid and use the direction of opening and vertical stretch factor information to determine the location of a few other points. Sketch the parabola. ( ) Verte: Direction of opening: Vertical stretch factor: From the verte, points are located: over unit, down units over units, down units over units, down units
Method : To sketch the graph of the function, determine the location of the verte. Then, we can determine the coordinates of another point on the graph. To do this, we can choose the -coordinate of a point and substitute this value of into the function to determine the corresponding -coordinate. ( ) Verte: (, ) Find another point: So, one point on the graph is (, ). For an point other than the verte, there is a corresponding point that is equidistant from the ais of smmetr. In this case, the corresponding point of (, ) is (, ). If other points are required, repeat the above process. Plot the verte, the additional points, and sketch the parabola. Verte Ais of Smmetr Direction of Opening Maimum or Minimum Value ( )
Eample : Sketch the graph of the function.5( ) and state the verte, ais of smmetr, direction of opening, maimum or minimum value, domain and range. Method : Determine the mapping rule. Substitute points from the table of values for the base function, mapping rule to obtain points for the new function. Sketch the parabola., into the Mapping Rule: 9 9.5( ) Method : Plot the verte on the grid and use the direction of opening and vertical stretch factor information to determine the location of a few other points. Sketch the parabola..5( ) Verte: Direction of opening: Vertical stretch factor: From the verte, points are located: over unit, up unit over units, up unit over units, up units
Method : To sketch the graph of the function, determine the location of the verte. Then, we can determine the coordinates of another point on the graph. To do this, we can choose the -coordinate of a point and substitute this value of into the function to determine the corresponding -coordinate..5( ) Verte: (, ) Find another point: So, one point on the graph is (, ). For an point other than the verte, there is a corresponding point that is equidistant from the ais of smmetr. In this case, the corresponding point of (, ) is (, ). If other points are required, repeat the above process. Plot the verte, the additional points, and sketch the parabola. Verte Ais of Smmetr Direction of Opening Maimum or Minimum Value.5( )
Eample : Without sketching the following functions, determine the verte, ais of smmetr, direction of opening, maimum or minimum value, domain, range, and the number of -intercepts, and mapping rule. Verte Ais of Smmetr Direction of Opening Maimum or Minimum Value Number of -intercepts Mapping Rule f ( ) ( 7) f ( ) ( 5) f ( ).5 f ( ) ( ) Eample : Determine a Quadratic Function in Verte Form Given its Graph Determine the quadratic function, in verte form, for the following graph: Solution: The verte of the graph is (, ) So, p = and q = The equation of the function in verte form a( p) q is Substitute the coordinates of an point from the graph (ecept the verte) into the equation to determine the value of a : a ( 5) The equation of the quadratic function is:
Eample : Model Problems Using Quadratic Functions in Verte Form The deck of the Lion s Gate Bridge in Vancouver is suspended from two main cables attached to the tops of two supporting towers. Between the towers, the main cables take the shape of a parabola as the support the weight of the deck. The towers are m tall relative to the water s surface and are 7 m apart. The lowest point of the cables is approimatel 67 m above the water s surface. a. Model the shape of the cables with a quadratic function in verte form. b. Determine the height above the water s surface of a point on the cables that is 9 m horizontall from either of the towers. Epress our answer to the nearest tenth of a metre. Solution: a. Sketch the parabolic shape of the cables on a set of coordinate aes. Choose a convenient point to represent the origin (, ) and use the information provided to label the coordinates of other ke points on the parabola. Determine the quadratic function that models the shape of the cables, where represents the height, in metres, above the water and represents the horizontal distance, in metres, from the left tower. b. A point on the cables that is 9 m horizontall from either of the towers would be at = m (9 m from the left tower) or = m (9 m from the right tower). Substitute either value of into the equation to determine height of this point above the water s surface: So, the height of this point above the water is approimatel m.