183k views
0 votes
Determine the following features of the quadratic function f(x) =-x2 -4x +5

Determine the following features of the quadratic function f(x) =-x2 -4x +5-example-1
User Ei Maung
by
8.0k points

1 Answer

5 votes

Solution

- The quadratic equation given is:


-x^2-4x+5=f(x)

Question 1:

- To know which direction a parabola opens, we follow the these rules:


\begin{gathered} \text{ Given,} \\ ax^2+bx+c \\ \\ \text{ if }a>0,\text{ the parabola opens Upwards} \\ \text{ If }a<0,\text{ the parabola opens downwards} \end{gathered}

- The value of a given is -1 < 0.

- Thus, the parabola opens downwards

Question 2:

- The y-intercept of the function is the coordinate of where the graph crosses the y-axis.

- In other words, this is also seen as the point where x = 0. Thus, substituting x = 0 into the equation given to us should readily reveal the y-value of the y-intercept.

- That is,


\begin{gathered} y=-x^2-4x+5 \\ put\text{ }x=0 \\ y=-0^2-4(0)+5 \\ y=5 \end{gathered}

- Thus, the y-intercept is (0, 5)

Question 3:

- The factorization of the function is given below:


\begin{gathered} f(x)=-x^2-4x+5 \\ \text{ We can rewrite the x-term as follows:} \\ -4x=-5x+x \\ \\ f(x)=-x^2-5x+x+5 \\ \text{ Thus, we can begin to factorize as follows:} \\ f(x)=-x(x+5)+1(x+5) \\ (x+5)\text{ is common so we can factorize again:} \\ \\ f(x)=(x+5)(1-x) \\ \\ f(x)=-1(x-1)(x+5) \end{gathered}

- Thus, the factorized form is:


f(x)=-1(x-1)(x+5)

Question 4:


\begin{gathered} \text{ The vertex of a parabola has its x-coordinate to be:} \\ x=-(b)/(2a)\text{ for the equation: }ax^2+bx+c \\ \\ \text{ Once we have the x-value, we can proceed to find the y-coordinate of the vertex by} \\ \text{ substituting this x-value into the equation.} \\ \text{ We have:} \\ \\ x=-(-4)/(2(-1))=-2 \\ \\ \\ y=-x^2-4x+5 \\ put\text{ }x=-2 \\ y=-(-2)^2-4(-2)+5 \\ y=-4+8+5 \\ y=9 \end{gathered}

- Thus, the vertex of the parabola is (-2, 9)

Question 5:

- The vertex form of the parabola is given by:


\begin{gathered} \text{ The formula is:} \\ y=a(x-h)^2+k \\ where, \\ (h,k)\text{ is the coordinate of the vertex.} \\ \\ \text{ The question has already given us }a=-1,\text{ thus, we can find the equation of the vertex as follows:} \\ h=-2,k=9\text{ \lparen Gotten from question 4\rparen} \\ \\ \therefore y=-1(x-(-2))^2+9 \\ \\ y=-1(x+2)^2+9 \end{gathered}

- The equation of the vertex is


y=-1(x+2)^2+9

User OmGanesh
by
8.6k points

No related questions found

Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories