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Quadratic functions are a type of polynomial that can be written in the form of $$ ax^2 + bx + c $$, where $$ a $$, $$ b $$, and $$ c $$ are constants, and $$ a eq 0 $$. 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, making it shaped like a U. If $$ a $$ is negative, the parabola opens downwards, resembling an inverted U.

The x-intercepts of a quadratic function are the points where the graph crosses the x-axis. To find these points, set the function equal to zero and solve for $$ x $$. This gives the equation:
$$ 2x^2 + 3x - 1 = 0 $$
Solve using methods such as factoring, completing the square, or applying the quadratic formula.
For the given function, we use the quadratic formula to obtain the x-intercepts. They are:
\[x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \]
Recalling the values $$ a = 2 $$, $$ b = 3 $$, and $$ c = -1 $$:
\[ x = \frac{-3 \, \pm \, \sqrt{(3)^2 - 4(2)(-1)}}{4} \]
This simplifies to:
\[ x = \frac{-3 \, \pm \, \sqrt{17}}{4} \]
Therefore, the x-intercepts of the function are:

• $$ x_1 = \frac{-3 + \sqrt{17}}{4} $$
• $$ x_2 = \frac{-3 - \sqrt{17}}{4} $$

The y-intercept is the point where the graph crosses the y-axis. This happens where $$ x = 0 $$. To find the y-intercept of a quadratic function, substitute 0 in place of $$ x $$ in the function $$ g(x) = 2x^2 + 3x - 1 $$.
Let's solve: \[ g(0) = 2(0)^2 + 3(0) - 1 = -1 \]
So, the y-intercept is the point (0, -1).
Notice that for any quadratic function, the y-intercept is always the constant term $$ c $$ in the function's equation.

The quadratic formula is a powerful tool for finding the x-intercepts of a quadratic function. It applies to any quadratic equation of the form $$ ax^2 + bx + c = 0 $$. The formula is:
\[x = \frac{-b \, \pm \, \sqrt{b^2 - 4ac}}{2a} \]

Here's how to use the quadratic formula step-by-step:

• Identify the coefficients $$ a $$, $$ b $$, and $$ c $$ in the quadratic equation.
• Plug these values into the formula.
• Calculate the discriminant, $$ b^2 - 4ac $$, which determines the nature of the roots (real or complex).
• Evaluate the expression under the square root.
• Simplify the results to find the values of $$ x $$.

In our given function, $$ 2x^2 + 3x - 1 $$, the coefficients are $$ a = 2 $$, $$ b = 3 $$, and $$ c = -1 $$. Plugging these into the quadratic formula gives the x-intercepts as: \[ x = \frac{-3 \, \pm \, \sqrt{17}}{4} \]
The quadratic formula always provides a reliable way to find the x-intercepts of any quadratic function.