To compare the graphs of different quadratic functions, let’s look at some important features:
Standard Form: Quadratic functions can be written as ( f(x) = ax^2 + bx + c ). Here, the number ( a ) shows if the graph goes up or down. If ( a ) is greater than 0, the graph opens up. If ( a ) is less than 0, the graph opens down.
Vertex: The vertex form is ( f(x) = a(x - h)^2 + k ). The vertex is the highest or lowest point of the graph, and you can find it at the point ( (h, k) ).
Axis of Symmetry: This is a line that goes through the vertex, written as ( x = h ). It divides the graph into two mirror images.
Roots: To find the x-intercepts—where the graph crosses the x-axis—you solve the equation ( ax^2 + bx + c = 0 ).
For example, if we compare the functions ( f(x) = x^2 - 2x + 1 ) and ( g(x) = -x^2 + 4 ), we can see that their vertices and intercepts are different. This helps us understand how each function behaves in its own way.
To compare the graphs of different quadratic functions, let’s look at some important features:
Standard Form: Quadratic functions can be written as ( f(x) = ax^2 + bx + c ). Here, the number ( a ) shows if the graph goes up or down. If ( a ) is greater than 0, the graph opens up. If ( a ) is less than 0, the graph opens down.
Vertex: The vertex form is ( f(x) = a(x - h)^2 + k ). The vertex is the highest or lowest point of the graph, and you can find it at the point ( (h, k) ).
Axis of Symmetry: This is a line that goes through the vertex, written as ( x = h ). It divides the graph into two mirror images.
Roots: To find the x-intercepts—where the graph crosses the x-axis—you solve the equation ( ax^2 + bx + c = 0 ).
For example, if we compare the functions ( f(x) = x^2 - 2x + 1 ) and ( g(x) = -x^2 + 4 ), we can see that their vertices and intercepts are different. This helps us understand how each function behaves in its own way.