I need to sketch the graph of t(x)=2x^2 +3x > or equal to 10x-x^2. How do I work out the points to plot on the graph?
one way: graph both functions and see where one is above the other. Visit
http://rechneronline.de/function-graphs/
and you can graph up to three functions at once. Set the y range for -20 to 20.
algebraic way:
2x^2+3x > 10x-x^2
3x^2 - 7x > 0
x(3x-7) > 0
either both factors are negative, or both are positive:
x is in (-∞,0) U (7/3,∞)
To sketch the graph of the inequality t(x) = 2x^2 + 3x ≥ 10x - x^2, you can follow these steps:
Step 1: Rewrite the inequality as an equation:
2x^2 + 3x = 10x - x^2
Step 2: Move all terms to one side of the equation to obtain a quadratic equation:
0 = 12x - 3x^2
Step 3: Rearrange the equation to its standard form:
3x^2 - 12x = 0
Step 4: Factor out the common factor, if possible:
3x(x - 4) = 0
Step 5: Set each factor equal to zero and solve for x:
3x = 0 or x - 4 = 0
Solving these equations gives you:
x = 0 or x = 4
These are the x-intercepts of the equation.
Step 6: Plot the x-intercepts on a graph. Mark a point at x = 0, and another point at x = 4 on the x-axis.
Step 7: Determine the behavior of the parabola between the x-intercepts. Since the coefficient of the x^2 term is positive, the parabola will open upwards.
Step 8: To find additional points to sketch the graph, choose any value of x within the intervals between the x-intercepts and evaluate t(x). For example, you can choose x = 1.
Substituting x = 1 into the equation t(x) = 2x^2 + 3x, we get:
t(1) = 2(1)^2 + 3(1) = 2 + 3 = 5
So, plot the point (1, 5) on the graph.
Step 9: Draw a smooth curve passing through all the plotted points. The part of the graph above the curve represents the solution to the inequality t(x) ≥ 10x - x^2.
Hence, the plotted graph represents the solution to the given inequality t(x) ≥ 10x - x^2.