Answer:
In layman's terms, the zero product property is simply a way of saying "anything times zero equals zero".
The x intercepts happen at (0.5, 0), and (-3, 0).
The vertex is a minimum.
The y-intercept happens at (0, -3).
For the last question, graphing f(x) can be done by taking a few key points (the vertex and x-intercepts are very good, as well as the vertex ± 1, 2, etc.). By that point you have enough data to sketch the curve. It helps to know in advance what the general shape of the function will be. In the case of a quadratic function, it will always be a parabola.
Step-by-step explanation:
Let's start by factoring the function:
f(x) = 2x² + 5x - 3
f(x) = 2x² - x + 6x - 3
f(x) = x(2x − 1) + 3 (2x−1)
f(x) = (x + 3)(2x - 1)
Now that we have it factored, finding its x intercept is quite easy. We just need to see which values of x would would give a y value of zero. That may sound backward, but note that when y is equal to zero, the graph is touching, or intercepting, the x-axis.
So if we say:
0 = (x + 3)(2x - 1)
then there are two solutions. First:
0 = (x + 3)(2x - 1)
0 / (x + 3) = (2x - 1)
0 = 2x - 1
2x = 1
x = 0.5
And the other:
0 = (x + 3)(2x - 1)
0 / (2x - 1) = (x + 3)
0 = x + 3
x = -3
So the x intercepts happen at (0.5, 0), and (-3, 0).
Next, regarding whether the vertex is a maximum or minimum, you can find this by the coefficient of the highest power of x in the function. If the coefficient is negative, then the vertex is a maximum. If it's positive, then it's a minimum. In this case the coefficient is 2, so the vertex is a minimum.
For part C, finding the y intercept is easy enough. Again, counter-intuitively the y intercept happens when x is equal to zero. So we just need to replace x with zero in the original equation to find it:
f(x) = 2x² + 5x - 3
f(0) = 2(0)² + 5(0) - 3
f(0) = -3
So the y-intercept happens at (0, -3).
