9.7 Graph Quadratic Functions Using Transformations
4 min read•june 25, 2024
Quadratic functions are all about parabolas. These U-shaped curves can shift up, down, left, or right. They can also stretch or compress. Understanding these helps you graph quadratics easily.
The , = a(x - h)^2 + k, is key. The 'a' controls direction and width, 'h' shifts horizontally, and 'k' shifts vertically. Mastering this form lets you quickly sketch any .
Graphing Quadratic Functions Using Transformations
Vertical shifts in quadratic functions
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Standard form of a quadratic function f(x)=a(x−h)2+k
a determines direction (opens upward if a>0, downward if a<0) and width (larger ∣a∣ results in narrower parabola)
h represents (right if h>0, left if h<0)
k represents (up if k>0, down if k<0)
Graphing steps:
Find by substituting x=h into function, vertex coordinates are (h,k)
Determine parabola direction based on sign of a
Apply vertical shift by moving graph up or down by k units (k>0 shifts up, k<0 shifts down by ∣k∣ units)
Examples:
f(x)=(x−2)2+3 has vertex at (2,3) and shifts up 3 units
g(x)=−(x+1)2−4 has vertex at (−1,−4) and shifts down 4 units
Horizontal shifts of quadratic graphs
Horizontal shifts move graph left or right by h units
h>0 shifts graph right by h units (x-coordinates increase)
h<0 shifts graph left by ∣h∣ units (x-coordinates decrease)
Sketching steps:
Identify vertex (h,k)
Plot vertex on coordinate plane
Determine parabola direction based on sign of a
Shift standard parabola y=x2 () horizontally by h units
Examples:
f(x)=(x−3)2 shifts right 3 units
g(x)=(x+2)2 shifts left 2 units
Stretching vs compressing quadratic functions
Value of a in standard form determines vertical stretching or compressing