Question

1. factor $2x^2 + 3x - 5$.

a

b y-int is

c the graph is (narrower/wider) than the parent function.
d the graph opens

Explanation:

Step1: Factor the quadratic

We use the AC method for factoring \(ax^2 + bx + c\). Here, \(a = 2\), \(b = 3\), \(c=-5\). \(AC=2\times(-5)= - 10\). We need two numbers that multiply to - 10 and add to 3. The numbers are 5 and - 2.
Rewrite the middle term: \(2x^{2}+5x - 2x-5\)
Group the terms: \((2x^{2}+5x)+(-2x - 5)=x(2x + 5)-1(2x + 5)=(2x + 5)(x - 1)\)

Step2: Find the y - intercept

The y - intercept of a function \(y = f(x)\) is found by setting \(x = 0\). For \(y=2x^{2}+3x - 5\), when \(x = 0\), \(y=2(0)^{2}+3(0)-5=-5\)

Step3: Compare the graph width with the parent function

The parent function of a quadratic is \(y=x^{2}\) (where \(a = 1\)). For the function \(y = ax^{2}+bx + c\), if \(|a|>1\), the graph is narrower than the parent function; if \(|a|<1\), it is wider. Here, \(a = 2\) and \(|2|>1\), so the graph is narrower.

Step4: Determine the direction the graph opens

For a quadratic function \(y=ax^{2}+bx + c\), if \(a>0\), the graph opens upward; if \(a < 0\), it opens downward. Here, \(a = 2>0\), so the graph opens upward.

Answer:

A. \((2x + 5)(x - 1)\)
B. \(-5\)
C. narrower
D. upward