$y = 5x^2 - 10x + 3$

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julian6821335 julian6821335

03-11-2026
Mathematics

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$y = 5x^2 - 10x + 3$

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T16:45:18-05:00

Identify the coefficients of the quadratic $Y = 5x^2 - 10x + 3$, which are $a = 5, b = -10, c = 3$.
Calculate the vertex coordinates using $x_v = -\frac{b}{2a}$ to get $x = 1$, and then $Y_v = -2$.
State the axis of symmetry as the vertical line passing through the vertex, $x = 1$.
Determine the range based on the minimum value and the direction of the opening: $[-2, \infty)$.
Final results: Vertex $\boxed{(1, -2)}$, axis of symmetry $\boxed{x = 1}$, and range $\boxed{[-2, \infty)}$.

Explanation

Analyzing the Quadratic Function

The given expression is a quadratic function of the form $Y = ax^2 + bx + c$, where $a = 5$, $b = -10$, and $c = 3$. Since the coefficient $a = 5$ is positive, the parabola opens upwards, which indicates that the function has a global minimum at its vertex. Our goal is to analyze this function by determining its vertex, axis of symmetry, and range.

Finding the Vertex

The $x$-coordinate of the vertex of a parabola defined by $ax^2 + bx + c$ is found using the formula $x_v = -\frac{b}{2a}$. Substituting the known values:
\[x_v = -\frac{-10}{2 \times 5} = \frac{10}{10} = 1.\]
Next, to find the $y$-coordinate of the vertex, we substitute $x_v = 1$ back into the original equation:
\[Y_v = 5(1)^2 - 10(1) + 3 = 5 - 10 + 3 = -2.\]
Therefore, the vertex of the parabola is at $(1, -2)$.

Axis of Symmetry and Vertex Form

The axis of symmetry for a parabola is the vertical line that passes through its vertex. Since the vertex has an $x$-coordinate of $1$, the equation for the axis of symmetry is $x = 1$. Additionally, we can express the function in vertex form $Y = a(x - h)^2 + k$, where $(h, k)$ is the vertex:
\[Y = 5(x - 1)^2 - 2.\]
This form clearly confirms our vertex location and the opening direction of the parabola.

Determining the Range and Final Conclusion

Since the parabola opens upwards ($a = 5 > 0$) and its minimum value is $-2$ at the vertex, the function takes all real values starting from $-2$ up to positive infinity. Thus, the range of the function is $[-2, \infty)$. Final summary of the analysis: the vertex is $(1, -2)$, the axis of symmetry is $x = 1$, and the range is $[-2, \infty)$.

Examples

Quadratic functions are fundamental in modeling real-world phenomena involving acceleration and gravity. For example, if you were to throw a ball into the air, its height over time could be described by a quadratic equation. By finding the vertex of such a function, you can determine the maximum height the ball reaches and the exact time it happens. In this case, analyzing $Y = 5x^2 - 10x + 3$ provides a foundation for understanding how to pinpoint critical points—like minimums or maximums—which is essential in fields ranging from engineering to economics, where optimizing resources is key.