- Identify the function \(f(x) = -2x^2 + x + 3\) as being in standard form.
- Note the leading coefficient \(a = -2 < 0\), which indicates the parabola opens down.
- Calculate the \(y\)-intercept by evaluating \(f(0)\), which yields \((0, 3)\).
- Determine the vertex and \(x\)-intercepts using the standard quadratic formulas, confirming that the other provided options are incorrect.
- The identified correct properties are: Standard form, Parabola opens down, and y-intercept: (0, 3).
Explanation
- Analyzing the Function Form and Basic Characteristics
The given quadratic function is \(f(x) = -2x^2 + x + 3\). This expression follows the form \(ax^2 + bx + c\), which is the standard form of a quadratic function, where \(a = -2\), \(b = 1\), and \(c = 3\). We will evaluate each of the provided characteristics based on this form.
- Evaluating Parabola Direction and y-intercept
First, consider the leading coefficient \(a = -2\). Since \(a < 0\), the parabola opens downward. Next, the \(y\)-intercept is found by evaluating the function at \(x=0\): \(f(0) = -2(0)^2 + 0 + 3 = 3\). Thus, the \(y\)-intercept is at the point \((0, 3)\). Both of these characteristics are correct.
- Calculating the Vertex Coordinates
The vertex \((h, k)\) of a standard quadratic function is given by \(h = -\frac{b}{2a}\) and \(k = f(h)\). Substituting the coefficients:
\(h = -\frac{1}{2(-2)} = -\frac{1}{-4} = 0.25\)
\(k = f(0.25) = -2(0.25)^2 + 0.25 + 3 = -2(0.0625) + 0.25 + 3 = -0.125 + 0.25 + 3 = 3.125\).
The vertex is \((0.25, 3.125)\), so the provided option of \((-2, 3)\) is incorrect.
- Calculating the x-intercepts
To find the \(x\)-intercepts, we solve \(f(x) = 0\):
\(-2x^2 + x + 3 = 0\)
Multiplying by \(-1\): \(2x^2 - x - 3 = 0\).
Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
\(x = \frac{1 \pm \sqrt{(-1)^2 - 4(2)(-3)}}{2(2)} = \frac{1 \pm \sqrt{1 + 24}}{4} = \frac{1 \pm \sqrt{25}}{4} = \frac{1 \pm 5}{4}\).
The roots are \(x_1 = \frac{6}{4} = 1.5\) and \(x_2 = \frac{-4}{4} = -1\).
The \(x\)-intercepts are \((1.5, 0)\) and \((-1, 0)\). The provided option \((3, 0)\) is incorrect.
- Conclusion of Correct Characteristics
Based on our analysis, the function is in standard form. The correct characteristics are:
- \(y\)-intercept: \((0, 3)\)
- Parabola opens down
- Standard form
All other provided characteristics are incorrect.
Examples
Understanding the standard form of a quadratic equation, \(f(x) = ax^2 + bx + c\), is crucial in physics for modeling projectile motion. For example, when you throw a ball into the air, its height over time follows this parabolic shape. By identifying the coefficients \(a, b,\) and \(c\), you can quickly determine the starting height (y-intercept at \(t=0\)) and whether the path is opening downward (meaning gravity is acting on the ball). Mastering these simple characteristics without needing a complex graph is a fundamental skill for analyzing real-world motion.