Let $$ f(x)=\frac{3 x^{2}-5 x+2}{2 x^{2}+5 x-25} $$ This function has: 1) Ay intercept at the point 2) $$ x $$ intercepts at the point(s) 3) Vertical asymptotes at $$ x=\square $$ 4) Horizontal asymptote at $$ y=\square $$

Question

UpStudy AI · Accepted Answer

To find the intercepts and asymptotes of the function $$ f(x)=\frac{3x^{2}-5x+2}{2x^{2}+5x-25} $$, we will follow these steps:

1. **y-intercept**: To find the y-intercept, we substitute $$ x=0 $$ into the function and solve for $$ y $$.
2. **x-intercepts**: To find the x-intercepts, we set $$ f(x) = 0 $$ and solve for $$ x $$.
3. **Vertical asymptotes**: Vertical asymptotes occur when the denominator of the function is equal to zero.
4. **Horizontal asymptote**: To find the horizontal asymptote, we compare the degrees of the numerator and denominator of the function.

Let's start by finding the y-intercept.
Evaluate the expression by following steps:
- step0: Evaluate:
  $$\frac{3x^{2}-5x+2}{2x^{2}+5x-25}$$
- step1: Substitute:
  $$\frac{3	imes 0^{2}-5	imes 0+2}{2	imes 0^{2}+5	imes 0-25}$$
- step2: Calculate:
  $$\frac{3	imes 0-5	imes 0+2}{2	imes 0^{2}+5	imes 0-25}$$
- step3: Calculate:
  $$\frac{3	imes 0-5	imes 0+2}{2	imes 0+5	imes 0-25}$$
- step4: Multiply:
  $$\frac{0-5	imes 0+2}{2	imes 0+5	imes 0-25}$$
- step5: Multiply:
  $$\frac{0+0+2}{2	imes 0+5	imes 0-25}$$
- step6: Multiply:
  $$\frac{0+0+2}{0+5	imes 0-25}$$
- step7: Multiply:
  $$\frac{0+0+2}{0+0-25}$$
- step8: Remove 0:
  $$\frac{2}{0+0-25}$$
- step9: Remove 0:
  $$\frac{2}{-25}$$
- step10: Rewrite the fraction:
  $$-\frac{2}{25}$$
The y-intercept of the function is at the point $$(-0.08, 0)$$.

Next, let's find the x-intercepts by setting $$ f(x) = 0 $$ and solving for $$ x $$.
Solve the equation by following steps:
- step0: Solve for $$x$$:
  $$\frac{3x^{2}-5x+2}{2x^{2}+5x-25}=0$$
- step1: Find the domain:
  $$\frac{3x^{2}-5x+2}{2x^{2}+5x-25}=0,x \in \left(-\infty,-5ight)\cup \left(-5,\frac{5}{2}ight)\cup \left(\frac{5}{2},+\inftyight)$$
- step2: Cross multiply:
  $$3x^{2}-5x+2=\left(2x^{2}+5x-25ight)	imes 0$$
- step3: Simplify the equation:
  $$3x^{2}-5x+2=0$$
- step4: Factor the expression:
  $$\left(x-1ight)\left(3x-2ight)=0$$
- step5: Separate into possible cases:
  $$\begin{align}&x-1=0\&3x-2=0\end{align}$$
- step6: Solve the equation:
  $$\begin{align}&x=1\&x=\frac{2}{3}\end{align}$$
- step7: Check if the solution is in the defined range:
  $$\begin{align}&x=1\&x=\frac{2}{3}\end{align},x \in \left(-\infty,-5ight)\cup \left(-5,\frac{5}{2}ight)\cup \left(\frac{5}{2},+\inftyight)$$
- step8: Find the intersection:
  $$\begin{align}&x=1\&x=\frac{2}{3}\end{align}$$
- step9: Rewrite:
  $$x_{1}=\frac{2}{3},x_{2}=1$$
The x-intercepts of the function are at the points $$(\frac{2}{3}, 0)$$ and $$(1, 0)$$.

Next, let's find the vertical asymptotes by setting the denominator of the function equal to zero and solving for $$ x $$.
Solve the quadratic equation by following steps:
- step0: Solve by factoring:
  $$2x^{2}+5x-25=0$$
- step1: Factor the expression:
  $$\left(x+5ight)\left(2x-5ight)=0$$
- step2: Separate into possible cases:
  $$\begin{align}&2x-5=0\&x+5=0\end{align}$$
- step3: Solve the equation:
  $$\begin{align}&x=\frac{5}{2}\&x=-5\end{align}$$
- step4: Rewrite:
  $$x_{1}=-5,x_{2}=\frac{5}{2}$$
The vertical asymptotes of the function occur at $$ x = -5 $$ and $$ x = 2.5 $$.

Finally, let's determine the horizontal asymptote by comparing the degrees of the numerator and denominator of the function. Since the degrees of the numerator and denominator are the same, the horizontal asymptote is the ratio of the leading coefficients of the numerator and denominator.

The horizontal asymptote is $$ y = \frac{3}{2} $$.

Therefore, the function has:
1. A y-intercept at the point $$(-0.08, 0)$$
2. X-intercepts at the points $$(\frac{2}{3}, 0)$$ and $$(1, 0)$$
3. Vertical asymptotes at $$ x = -5 $$ and $$ x = 2.5 $$
4. A horizontal asymptote at $$ y = \frac{3}{2} $$
1. y-intercept at $$(-0.08, 0)$$ 2. x-intercepts at $$(\frac{2}{3}, 0)$$ and $$(1, 0)$$ 3. Vertical asymptotes at $$ x = -5 $$ and $$ x = 2.5 $$ 4. Horizontal asymptote at $$ y = \frac{3}{2} $$

Let
This function has:

Ay intercept at the point

intercepts at the point(s)

Vertical asymptotes at

Horizontal asymptote at

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Pre Calculus Questions

Let This function has: Ay intercept at the point intercepts at the point(s) Vertical asymptotes at Horizontal asymptote at