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=)

Transcript text: 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: $y=$ $\square$ $a=$ $\square$ $h=$ $\square$ $k=$ $\square$

Solution

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.

Step 1: Convert to Vertex Form

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

\[ y = a(x - h)^2 + k \]

Step 2: Identify Coefficients

From the equation, we identify the coefficients:

$ a = 4 $
$ b = -7 $
$ c = 5 $

Step 3: Calculate $ h $ and $ k $

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

Step 4: Write the Vertex Form

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

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