Question

part 1:
convert the quadratic equation below into both intercept form and vertex form. show your work! state the key feature each form tells you. based on the key features you find from each form, sketch a graph on the grid below:
standard form
$y = x^2 - 10x + 21 = 0$
$y = ax^2 + bx + c$
key feature:
$y$-intercept
$(3, 0)$
intercept form
$y = \underline{\quad\quad\quad\quad\quad\quad\quad}$
key features:
roots/$x$-int $(m, 0)$
vertex form
$y = \underline{\quad\quad\quad\quad\quad\quad\quad}$
key features:
vertex $(h, k)$
sketch:
graph with grid, labeled points $(3, 0)$, $(7, 0)$, $(5, -4)$, and regions labeled \positive\ and
egative\
based on your sketch determine the following key features:
domain: $(-\infty, \infty)$ all real #s
range: $y \geq -4$? (handwritten as $y \leq 7$? $(7, 0)$?)
interval of increase
interval where graph is negative
end behavior as $x$ increases, $y$ $\underline{\quad\quad\quad\quad}$
as $x$ decreases, $y$ $\underline{\quad\quad\quad\quad}$

Explanation:

Step1: Find Intercept Form (Factor the quadratic)

To convert \( y = x^2 - 10x + 21 \) to intercept form (\( y=(x - r_1)(x - r_2) \)), we factor the quadratic. We need two numbers that multiply to \( 21 \) and add to \( -10 \). Those numbers are \( -3 \) and \( -7 \). So, \( x^2 - 10x + 21=(x - 3)(x - 7) \). Thus, intercept form is \( y=(x - 3)(x - 7) \).

Step2: Find Vertex Form (Complete the square)

For vertex form \( y=a(x - h)^2 + k \), start with \( y = x^2 - 10x + 21 \).

1. Group \( x \)-terms: \( y=(x^2 - 10x)+21 \).
2. Complete the square for \( x^2 - 10x \): take half of \( -10 \) (which is \( -5 \)), square it (\( 25 \)), add and subtract it: \( y=(x^2 - 10x + 25 - 25)+21 \).
3. Rewrite as a perfect square and simplify: \( y=(x - 5)^2 - 25 + 21=(x - 5)^2 - 4 \). So vertex form is \( y=(x - 5)^2 - 4 \).

Answer:

• Intercept Form: \( y=(x - 3)(x - 7) \)
• Vertex Form: \( y=(x - 5)^2 - 4 \)