Questions: Put the equation (y=4 x^2-7 x+5) in vertex form and state the values of (a, h) and (k). Enter the exact answers. Vertex form: (y=) (a=) (h=) (k=)

Solution Steps

To convert the quadratic equation $y = 4x^2 - 7x + 5$ into vertex form, we need to complete the square. The vertex form of a quadratic equation is $y = a(x-h)^2 + k$, where $(h, k)$ is the vertex of the parabola. We will first factor out the coefficient of $x^2$ from the first two terms, then complete the square inside the parentheses, and finally adjust the constant term to maintain equality.

To convert the quadratic equation $y = 4x^2 - 7x + 5$ into vertex form, we complete the square. The vertex form is given by:

$$y = a(x - h)^2 + k$$

From the equation, we identify the coefficients:

• $a = 4$
• $b = -7$
• $c = 5$

Using the formulas for $h$ and $k$: $$h = -\frac{b}{2a} = -\frac{-7}{2 \cdot 4} = \frac{7}{8} = 0.875$$ $$k = c - \frac{b^2}{4a} = 5 - \frac{(-7)^2}{4 \cdot 4} = 5 - \frac{49}{16} = 5 - 3.0625 = 1.9375$$

Substituting $h$ and $k$ back into the vertex form, we have:

$$y = 4\left(x - 0.875\right)^2 + 1.9375$$

Final Answer