Graphing Quadratics: Vertex and Intercept Form
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1 Algebra : UNIT Graphing Quadratics: Verte and Intercept Form Date: Welcome to our second function famil...the QUADRATIC FUNCTION! f() = (the parent function) What is different between this function and the absolute value function? ALL quadratic functions have ke features that we care about:. Verte a point. Ais of smmetr Equation of a line 5. Min or ma A point. -intercept(s) A point(s) 5. -intercept A point 7. End behavior As, f() As, f(). Domain An interval. Increasing and Decreasing Intervals An interval 9. Range An interval
2 Algebra : UNIT Verte Form: = a ( h) + k Eplain the difference between an absolute value function and a quadratic function when ou are looking at the equations: Given the quadratic function = a ( - h) + k *NOTE: a is NOT slope like it was for absolute value! If a > 0, does the graph open up or down? If a < 0, does the graph open up or down? If a >, does the graph have a vertical stretch or vertical shrink? If 0 < a <, does the graph have a vertical stretch or vertical shrink? What is the verte? What does the parameter k control? What does the parameter h control? EXAMPLE: Write an equation of a quadratic function with a verte at (-, 5) that opens down and has vertical shrink. Complete the table below WITHOUT our calculator: Function Direction Dilation Verte Domain Range Stretch = ( + ) Up + Shrink Down Standard = ( ) + 5 = ( + ) = Up Down Up Down Up Down How can ou tell if a verte is a ma or min without graphing? How did we find stretch or shrink for absolute value? Stretch Shrink Standard Stretch Shrink Standard Stretch Shrink Standard Wh can't we use a as slope for quadratic functions?
3 Algebra : UNIT Graphing in Verte From WITHOUT a Calculator:. Find our verte, (h,k).. Note the value of a in our function. Find and plot other points b making horizontal (h) and vertical (v) moves from the verte. Draw a smooth curve through our plotted points. Horizontal Vertical a a 9a a 5 5a. =. = Verte: Stretch or shrink? H V 0 0 Verte: Stretch or Shrink? H V 0 0. = ( 5) 7. = ( + ) + Verte: Stretch or shrink? 0 Verte: Stretch or shrink? 0 H V 0 H V 0
4 Algebra : UNIT Writing the Equation of a Graphed Function in Verte Form:. What is the general form of the parent graph in verte form?. Put (h, k) the verte into our equation.. Substitute a different point into the equation for and. Solve for a.. Write the final equation using a from step. EXAMPLE: Given the graph, write the quadratic equation in verte form for each of the following without using a calculator:.. 7 The verte is. The verte is. Equation: Equation:
5 Algebra : UNIT Intercept Form: = a( p)( q) Quick review: State whether each statement is alwas, sometimes, or never true.. The graph of a quadratic function is a V shape.. The range of a quadratic function is the set of all real numbers.. The graph of a quadratic function contains the point (0, 0).. The verte of a parabola occurs at the minimum value of the function. 5. A quadratic function has two real solutions.. If a quadratic function s verte is on the -ais, then it has eactl one solution. 7. The inverse of a quadratic function is also a function. Is this reall a quadratic? Graph these with our calculator and see.. = ( + )( - ) (p =, q = ) Verif algebraicall b multipling (FOIL): 0 0. = ( - )( - ) (p =, q = ) Verif algebraicall (FOIL): What patterns do ou notice in this equation tpe? 0 0 What is nice about INTERCEPT form? What was nice about VERTEX form? 5
6 Algebra : UNIT Graphing in Intercept Form WITHOUT a Calculator. Find and graph the -intercepts (zero product propert ) or p and q with OPPOSITE signs!. Find and graph the verte (ais of smmetr ). The mid-point of the -intercepts: o = pp+qq to find -coordinate. o Plug into original equation to find -coordinate.. Connect the points to make the parabola.. = ( )( ). ff() = ( + )( ) -intercepts: -intercepts: Verte: Verte: Domain: Range: -intercept: Domain: Range: -intercept: Increasing Interval: Decreasing Interval: Ma or min? As, ff() As, ff() Increasing Interval: Decreasing Interval: Ma or min? As, ff() As, ff()
7 Algebra : UNIT. = ( ). = ( )( + ) -intercepts: -intercepts: Verte: Verte: Domain: Range: -intercept: Domain: Range: -intercept: Increasing Interval: Decreasing Interval: Ma or min? As, ff() As, ff() Increasing Interval: Decreasing Interval: Ma or min? As, ff() As, ff() What happens if we give # vertical stretch or shrink? New quadratic in intercept form Did changing "a" affect the - or -intercepts? 7
8 Algebra : UNIT Writing the Equation of a Graphed Function in Intercept Form:. What is the general form of the parent graph in intercept form?. Put p and q (the -intercepts) into our equation. *Remember: OPPOSITE sign!. Substitute a different point into the equation for and. Solve for a.. Write the final equation using a from step. EXAMPLE: Given the graph, write the quadratic equation in intercept form for each of the following without using a calculator: The verte is. The verte is. Equation: Equation:
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