Quadratic graphs are visual representations of quadratic functions, showcasing parabolas that open upwards or downwards. The vertex form, f(x)=a(x-h)^2+k, is crucial for graphing, revealing the vertex and the parabola's width. This text delves into plotting techniques, translating and scaling parabolas, and interpreting quadratic inequalities. It also guides on sketching quadratic functions and graphing inequalities, as well as deriving equations from graphs.
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Quadratic functions are second-degree polynomials with the highest power of the variable being 2
Shape and orientation
Parabolas are symmetrical curves that open either upwards or downwards, with their shape and orientation determined by the coefficient a
Vertex form
The vertex form of a quadratic function, f(x)=a(x-h)^2+k, directly reveals the parabola's vertex and direction of opening
The parameters a, h, and k in the vertex form equation determine the parabola's width, position, and response to horizontal and vertical shifts
The vertex, direction of opening, and width can be determined from the vertex form equation, making it easier to graph a quadratic function
The values of h and k in the vertex form equation determine the horizontal and vertical shifts of the parabola, respectively
The coefficient a in the vertex form equation determines the vertical scaling or dilation of the parabola
To graph a quadratic function, identify the vertex, direction of opening, and intercepts, and plot them on the graph
Quadratic inequalities are expressed using inequality symbols and are represented by a parabola on a graph
The shaded region on the graph denotes the set of points that satisfy the inequality, and the line style (solid or dashed) indicates whether the inequality is inclusive or exclusive
To graph a quadratic inequality, sketch the corresponding quadratic function and use the inequality symbol and line style to determine the shaded region on the graph
The vertex and another point on the parabola can be used to calculate the coefficients a, h, and k in the vertex form equation
The direction of the shaded region and the line style on the graph can be used to determine the appropriate inequality symbol for a quadratic inequality, or the equation can be directly formulated from the graph
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Vertex form of a quadratic function
f(x)=a(x-h)^2+k, where a, h, k are constants determining shape and position.
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Effect of coefficient 'a' on a parabola
Coefficient 'a' determines width and orientation; positive 'a' opens upward, negative 'a' opens downward.
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Meaning of parameters 'h' and 'k' in vertex form
'h' and 'k' indicate the parabola's vertex, 'h' is the x-coordinate, 'k' is the y-coordinate.
