Sketching the Graph of ( f(x) 5x^2 - 4x - 3 ): A Step-by-Step Guide

Graph sketching of a quadratic function ( f(x) 5x^2 - 4x - 3 ) involves several key steps that assist in visualizing the parabola accurately. This guide will walk you through the process, starting from understanding the function's form and progressing to identifying key points and drawing the graph.

1. Understand the Function Form

The given function is ( f(x) 5x^2 - 4x - 3 ), which is a quadratic function in standard form ( f(x) ax^2 bx c ). Here, ( a 5 ), ( b -4 ), and ( c -3 ). Identifying these coefficients helps in finding the axis of symmetry, the vertex, and the intercepts on the coordinate plane.

2. Convert to Vertex Form

Converting the quadratic function to vertex form ( f(x) a(x - h)^2 k ) simplifies the process of identifying the vertex. This form directly reveals the axis of symmetry and the minimum (or maximum) value of the function. To convert, complete the square:

[ f(x) 5x^2 - 4x - 3 ]

Factor out the coefficient of ( x^2 ) from the first two terms:

[ f(x) 5(x^2 - frac{4}{5}x) - 3 ]

Complete the square by adding and subtracting the square of half the coefficient of ( x ), which is ( left( frac{-4/5}{2} right)^2 left( -frac{2}{5} right)^2 frac{4}{25} ).

[ f(x) 5 left( x^2 - frac{4}{5}x frac{4}{25} - frac{4}{25} right) - 3 ]

[ f(x) 5 left( x^2 - frac{4}{5}x frac{4}{25} right) - 5 cdot frac{4}{25} - 3 ]

[ f(x) 5 left( x - frac{2}{5} right)^2 - frac{4}{5} - 3 ]

[ f(x) 5 left( x - frac{2}{5} right)^2 - frac{19}{5} ]

The vertex form is thus ( f(x) 5 left( x - frac{2}{5} right)^2 - frac{19}{5} ).

From this, the vertex ((h, k)) is (left( frac{2}{5}, -frac{19}{5} right)). The axis of symmetry is ( x frac{2}{5} ), and the minimum value of the function is ( -frac{19}{5} ) at ( x frac{2}{5} ).

3. Calculate Key Points

Mark the x-axis and y-axis on a Cartesian plane, ensuring an appropriate scale. Calculate key points by choosing several x-values and substituting them into the original or vertex form of the function.

Choose ( x 0 ):[ f(0) 5(0)^2 - 4(0) - 3 -3 ]Point: ((0, -3))Choose ( x 1 ):[ f(1) 5(1)^2 - 4(1) - 3 5 - 4 - 3 -2 ]Point: ((1, -2))Choose ( x -1 ):[ f(-1) 5(-1)^2 - 4(-1) - 3 5 4 - 3 6 ]Point: ((-1, 6))Choose ( x frac{2}{5} ):[ fleft( frac{2}{5} right) -frac{19}{5} ]Point: (left( frac{2}{5}, -frac{19}{5} right))

Plot these points and draw a smooth curve through them, using the axis of symmetry as a guide. The curve should open upwards (since ( a 5 > 0 )).

4. Draw the Graph

Based on the calculations and the vertex form, you can now draw the graph. The x-intercepts can be found by setting ( f(x) 0 ) and solving the quadratic equation:

[ 5x^2 - 4x - 3 0 ]

Using the quadratic formula ( x frac{-b pm sqrt{b^2 - 4ac}}{2a} ):

[ x frac{-(-4) pm sqrt{(-4)^2 - 4 cdot 5 cdot (-3)}}{2 cdot 5} ]

[ x frac{4 pm sqrt{16 60}}{10} ]

[ x frac{4 pm sqrt{76}}{10} ]

[ x frac{4 pm 2sqrt{19}}{10} ]

[ x frac{2 pm sqrt{19}}{5} ]

The x-intercepts are approximately ( x frac{2 sqrt{19}}{5} ) and ( x frac{2 - sqrt{19}}{5} ).

Add these x-intercepts to your graph, and connect the points smoothly to form the parabola.

By following these steps, you can accurately sketch the graph of ( f(x) 5x^2 - 4x - 3 ) and have a comprehensive understanding of its behavior.

Conclusion

Graph sketching of ( f(x) 5x^2 - 4x - 3 ) involves converting the standard form to vertex form for easier identification of key points, calculating key points, and then plotting and connecting the points. Following these steps ensures an accurate and insightful visual representation of the quadratic function.