Understanding Quadratic Functions and Graphs
Explore the world of quadratic functions and graphs, including their forms, properties, and visual representations through parabolas. Learn about symmetries, axes of symmetry, vertices, and intercepts of quadratic functions. Discover the similarities and differences in various quadratic functions graphed in this informative content.
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6 July 2025 The quadratic function of the form f (x) = ax2 + bx c LO: Draw the graph of quadratic functions of the form y= ax2 + bx + c www.mathssupport.org www.mathssupport.org
Function of the form y = ax2 + bx + c or f(x) = ax2 + bx + c The simplest quadratic function is y = x2 y The shape of this graph is called a parabola. 10 9 8 7 6 The graph has a minimum at the point (0, 0). 5 4 3 2 1 x -10 1 2 3 4 5 6 7 9 10 -9 -8 -7 -6 -5 -4 -3 -2 8 -1 0 -1 The graph is symmetrical about the y-axis. -2 -3 -4 -5 -6 -7 -8 -9 -10 www.mathssupport.org www.mathssupport.org
Function of the form y = ax2 + bx + c or f(x) = ax2 + bx + c If you look at the graphs of other quadratic functions, you should notice some similarities. y y = x2 + 2x - 1 10 This graph has a curve known as a parabola. The graph has a minimum called a vertex. The graph is symmetrical on the left and right side of this vertical line This imaginary vertical line is called axis of symmetry. y-intercept(0, -1) 9 8 7 6 5 4 3 2 1 x -10 1 2 3 4 5 6 7 9 10 -9 -8 -7 -6 -5 -4 -3 -2 8 -1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 www.mathssupport.org www.mathssupport.org
Function of the form y = ax2 + bx + c or f(x) = ax2 + bx + c If you look at the graphs of other quadratic functions, you should notice some similarities. y y = 3x2 - 4x + 2 This graph has a curve known as a parabola. The graph has a minimum called a vertex. The graph is symmetrical on the left and right side of this vertical line This imaginary vertical line is called axis of symmetry. y-intercept(0, 2) 10 9 8 7 6 5 4 3 2 1 x -10 1 2 3 4 5 6 7 9 10 -9 -8 -7 -6 -5 -4 -3 -2 8 -1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 www.mathssupport.org www.mathssupport.org
Function of the form y = ax2 + bx + c or f(x) = ax2 + bx + c If you look at the graphs of other quadratic functions, you should notice some similarities. y y = -2x2 + 2x + 3 This graph has a curve known as a parabola. The graph has a maximum called a vertex. The graph is symmetrical on the left and right side of this vertical line This imaginary vertical line is called axis of symmetry. y-intercept(0, 3) 10 9 8 7 6 5 4 3 2 1 x -10 1 2 3 4 5 6 7 9 10 -9 -8 -7 -6 -5 -4 -3 -2 8 -1 0 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 www.mathssupport.org www.mathssupport.org
Function of the form y = ax2 + bx + c or f(x) = ax2 + bx + c For quadratic functions in standard form y = ax2 + bx + c the graph crosses the y-axis at (0, c) . y y = x2 + 2x - 1 10 9 This graph crosses the y-axis at (0, -1). 8 7 6 5 The equation of the axis of symmetry is 4 3 2 x =-b 1 x -10 1 2 3 4 5 6 7 9 10 -9 -8 -7 -6 -5 -4 -3 -2 8 -1 2a 0 -1 x =-2 -2 -3 2 -4 x = -1 -5 -6 -7 -8 -9 -10 www.mathssupport.org www.mathssupport.org
Function of the form y = ax2 + bx + c or f(x) = ax2 + bx + c For quadratic functions in standard form y = ax2 + bx + c the graph crosses the y-axis at (0, c) . y y = 3x2 - 4x + 2 This graph crosses the y-axis at (0, 2). 10 9 8 7 6 5 The equation of the axis of symmetry is 4 3 2 x =-b 1 x -10 1 2 3 4 5 6 7 9 10 -9 -8 -7 -6 -5 -4 -3 -2 8 -1 0 -1 2a x =4 -2 -3 6 -4 x =2 -5 -6 3 -7 -8 -9 -10 www.mathssupport.org www.mathssupport.org
Function of the form y = ax2 + bx + c or f(x) = ax2 + bx + c For quadratic functions in standard form y = ax2 + bx + c the graph crosses the y-axis at (0, c) . y y = -2x2 + 2x + 3 10 9 This graph crosses the y-axis at (0, 3). 8 7 6 5 The equation of the axis of symmetry is 4 3 2 x =-b 1 x -10 1 2 3 4 5 6 7 9 10 -9 -8 -7 -6 -5 -4 -3 -2 8 -1 2a 0 -1 x =-2 -2 -3 -4 -4 x =1 -5 -6 2 -7 -8 -9 -10 www.mathssupport.org www.mathssupport.org
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