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Complete Table 3 for the equation y = x³ + 4x² + x − 3. Use the scale of 2 cm to represent 1 unit on the x-axis and 2cm to represent 5 units on the y-axis. Draw the graph of y = x³ + 4x² + x − 3. Use your graph to solve the equation x³ + 4x² + 3/2x − 4 = 0.

Mathematics

ScanSolve AI Answer

Graph of y = x³ + 4x² + x − 3

Use the left and right arrow keys to trace the curve, up and down to jump between key points, and plus and minus to zoom.

Key points of y = x³ + 4x² + x − 3
Pointxy
x-intercept−3.460
Maximum−2.543.88
x-intercept−1.240
Minimum−0.13−3.06
y-intercept0−3
x-intercept0.70

Step-by-step solution

3 steps

  1. Complete Table 1 for y=x2−3x−4y = x^2 - 3x - 4

    For x=−1x = -1:

    y=(−1)2−3(−1)−4y = (-1)^2 - 3(-1) - 4 y=1+3−4y = 1 + 3 - 4 y=0y = 0

    For x=0x = 0:

    y=(0)2−3(0)−4y = (0)^2 - 3(0) - 4 y=0−0−4y = 0 - 0 - 4 y=−4y = -4

    For x=1x = 1:

    y=(1)2−3(1)−4y = (1)^2 - 3(1) - 4 y=1−3−4y = 1 - 3 - 4 y=−6y = -6

    For x=2x = 2:

    y=(2)2−3(2)−4y = (2)^2 - 3(2) - 4 y=4−6−4y = 4 - 6 - 4 y=−6y = -6

    For x=3x = 3:

    y=(3)2−3(3)−4y = (3)^2 - 3(3) - 4 y=9−9−4y = 9 - 9 - 4 y=−4y = -4

    Completed table:

    x−10123y0−4−6−6−4\begin{array}{c|c|c|c|c|c} x & -1 & 0 & 1 & 2 & 3 \\ \hline y & 0 & -4 & -6 & -6 & -4 \\ \end{array}

  2. Draw the graph

    Use the scale: 1 cm represents 1 unit on the x-axis and 1 cm represents 1 unit on the y-axis.
    Plot points: (−1,0)(-1, 0), (0,−4)(0, -4), (1,−6)(1, -6), (2,−6)(2, -6), (3,−4)(3, -4).
    Join with a smooth curve. The graph is a parabola opening upwards (vertex near (1.5,−6.25)(1.5, -6.25)).

  3. Use graph to solve x2−3x−4=0x^2 - 3x - 4 = 0

    The graph crosses y = 0 at x=−1x = -1 (exact) and at x=4x = 4 (by extending the curve beyond x = 3).

    Solutions: x=−1, 4\mathbf{x = -1,\ 4}
    x = -1,\ 4

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