How To Graph A Quadratic Function And Find Its Vertex And Intercepts

Graph of a quadratic function

f(x) = x

A quadratic function is a polynomial function with degree two. The general form of a quadratic function is:

f(x) = ax^2 + bx + c

where a, b, and c are real numbers and a cannot be equal to zero. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of a.

If a is positive, the parabola opens upwards, and if a is negative, the parabola opens downwards. The vertex of the parabola occurs at the point (-b/2a , f(-b/2a)), and the axis of symmetry of the parabola is the vertical line x = -b/2a.

To graph a quadratic function, we can use the following steps:

1. Find the vertex of the parabola by using the formula (-b/2a , f(-b/2a)).
2. Find the y-intercept of the parabola by setting x=0 in the equation and solving for y.
3. Find the x-intercepts of the parabola by setting y=0 in the equation and solving for x using the quadratic formula.
4. Plot the vertex, y-intercept, and x-intercepts on the coordinate plane.
5. Determine the direction of the parabola’s opening:
– If a is positive, the parabola opens upwards.
– If a is negative, the parabola opens downwards.
6. Sketch the parabola through these points, making sure to show the axis of symmetry.

Here is an example of how to graph the quadratic function f(x) = x^2 – 4x + 3:

1. Find the vertex of the parabola:
– b = -4 and a = 1
– x-coordinate of the vertex is -b/2a = 2
– y-coordinate of the vertex is f(2) = (2)^2 – 4(2) +3 = -1
– Therefore, the vertex is (2,-1)

2. Find the y-intercept of the parabola:
– Set x=0 in the equation: f(0) = 0^2 – 4(0) + 3 = 3
– Therefore, the y-intercept is (0,3)

3. Find the x-intercepts of the parabola:
– Set y=0 in the equation: x^2 – 4x + 3 = 0
– Use the quadratic formula to solve for x:
x = (4 ± sqrt(16 – 4(1)(3))) / 2(1)
x = (4 ± sqrt(4)) / 2
x = 2 ±1
Therefore, the x-intercepts are (1,0) and (3,0)

4. Plot the vertex, y-intercept, and x-intercepts on the coordinate plane:

5. Determine the direction of the parabola’s opening:
– a is positive, so the parabola opens upwards.

6. Sketch the parabola through these points, making sure to show the axis of symmetry:

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