The vertex form of the function $y=x^2+8 x-3$ is $y=(x+$ □ $)^2+$ □ .

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03-11-2026
Mathematics

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The vertex form of the function $y=x^2+8 x-3$ is $y=(x+$ □ $)^2+$ □ .

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GinnyAnswer GinnyAnswer · Answer 1 · 2026-11-03T13:47:46-05:00

To convert $y = x^2 + 8x - 3$ into vertex form, identify the coefficient of $x$, which is 8.
Take half of this coefficient (4) and square it (16) to complete the square: $x^2 + 8x + 16 - 16 - 3$.
Factor the expression as a perfect square: $(x+4)^2 - 19$.
The final vertex form is $y = (x + \boxed{4})^2 + \boxed{-19}$.

Explanation

Understanding the Problem

The problem asks to rewrite the quadratic function $y = x^2 + 8x - 3$ in the vertex form $y = (x + h)^2 + k$. The standard form of a quadratic function is $y = ax^2 + bx + c$. In our case, $a = 1$, $b = 8$, and $c = -3$. To convert this to vertex form, we use a method known as completing the square. The vertex form $y = a(x - h)^2 + k$ allows us to easily identify the vertex of the parabola, $(h, k)$.

Completing the Square

To complete the square, we focus on the first two terms: $x^2 + 8x$. We want to add a constant to these terms to form a perfect square trinomial of the form $x^2 + bx + (\frac{b}{2})^2 = (x + \frac{b}{2})^2$. Here, the coefficient of $x$ is $8$. We take half of this coefficient, which is $\frac{8}{2} = 4$, and then square it to get $4^2 = 16$. To keep the expression equivalent to the original one, we add and subtract this value:
\[x^2 + 8x - 3 = (x^2 + 8x + 16) - 16 - 3\]

Finalizing the Vertex Form

Now, we factor the perfect square trinomial $(x^2 + 8x + 16)$ into $(x + 4)^2$. Substituting this back into our expression, we get:
\[(x^2 + 8x + 16) - 16 - 3 = (x + 4)^2 - 19\]
Thus, the function in vertex form is $y = (x + 4)^2 - 19$. Comparing this to the desired form $y = (x + \boxed{\phantom{0}})^2 + \boxed{\phantom{0}}$, we identify the missing values as $4$ and $-19$.

Examples

The vertex form is incredibly useful in real-world scenarios such as projectile motion. For example, if a ball is thrown into the air, its height over time is often modeled by a quadratic function. By rewriting this function in vertex form, we can immediately identify the maximum height the ball reaches (the $k$ value) and the time at which it happens (the $h$ value). Understanding how to convert between forms allows us to quickly analyze the peak of a trajectory, making physics and engineering problems much simpler to solve!