Math Problem Statement

What is the y-value of the vertex of the quadratic function: -7x^2 - 6x + 10?

Solution

The quadratic function provided is:

f(x)=7x26x+10f(x) = -7x^2 - 6x + 10

To find the y-value of the vertex of this quadratic, we can follow these steps:

1. Find the x-coordinate of the vertex:

The x-coordinate of the vertex for a quadratic equation ax2+bx+cax^2 + bx + c can be found using the formula:

x=b2ax = \frac{-b}{2a}

In this case, a=7a = -7 and b=6b = -6. Plugging these values into the formula:

x=(6)2(7)=614=37x = \frac{-(-6)}{2(-7)} = \frac{6}{-14} = -\frac{3}{7}

2. Find the y-value (f(x)) at the vertex:

Now substitute x=37x = -\frac{3}{7} into the function to find the y-value:

f(37)=7(37)26(37)+10f\left(-\frac{3}{7}\right) = -7\left(-\frac{3}{7}\right)^2 - 6\left(-\frac{3}{7}\right) + 10

Let's compute this step-by-step:

f(37)=7(949)+6(37)+10=6349+187+10f\left(-\frac{3}{7}\right) = -7\left(\frac{9}{49}\right) + 6\left(\frac{3}{7}\right) + 10 = -\frac{63}{49} + \frac{18}{7} + 10

Simplify further:

f(37)=6349+12649+49049f\left(-\frac{3}{7}\right) = -\frac{63}{49} + \frac{126}{49} + \frac{490}{49}

f(37)=63+126+49049=55349f\left(-\frac{3}{7}\right) = \frac{-63 + 126 + 490}{49} = \frac{553}{49}

Thus, the y-value of the vertex is:

5534911.29\boxed{\frac{553}{49}} \approx 11.29

Would you like more details or have any questions?

Here are 5 related questions:

  1. How do you find the vertex of a quadratic equation in vertex form?
  2. What does the vertex of a parabola represent in real-world contexts?
  3. How do you determine whether the vertex is a maximum or minimum?
  4. Can the quadratic formula be used to find the vertex?
  5. How does changing the coefficient of x2x^2 affect the shape of the parabola?

Tip: For any quadratic equation, the vertex formula is a quick way to find the turning point of the parabola.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Vertex of a Parabola

Formulas

x = -b / 2a
y = f(x) for the vertex

Theorems

Vertex Formula for a Quadratic Equation

Suitable Grade Level

Grades 8-10