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besties this is 100 points & i'll give brainliest istg i need to know this
Find the unknown values of b and c in the equation y = -x² + bx + c given its vertex is (-7, -5).

Respuesta :

Nayefx

Answer:

[tex] \displaystyle b = - 14[/tex]

[tex] \displaystyle c = - 54[/tex]

Step-by-step explanation:

to figure out b we can consider the following formula:

[tex] \displaystyle \frac{ - b}{2a} = h[/tex]

the form of vertex coordinate given by

[tex] \displaystyle (h,k)[/tex]

so according to the question

[tex] \displaystyle (h,k) = ( - 7, - 5)[/tex]

by order pair we obtain:

[tex] \displaystyle h = - 7, k = - 5[/tex]

now substitute the value of h and a to the formula:

[tex] \displaystyle \frac{ - b}{2(- 1)} = - 7[/tex]

simplify multiplication:

[tex] \displaystyle \frac{ - b}{ - 2} = - 7[/tex]

cross multiplication:

[tex] \displaystyle - b = 14[/tex]

divide both sides by -1:

[tex] \displaystyle b = - 14[/tex]

now we need to figure out C to do so substitute -7 for x and -5 for y which yields:

[tex] \displaystyle - {7}^{2} - 14( - 7) + c = - 5[/tex]

simplify square:

[tex] \displaystyle -49 - 14( - 7) + c = - 5[/tex]

simplify multiplication:

[tex] \displaystyle -49 + 98+ c = - 5[/tex]

simplify addition:

[tex] \displaystyle 49+ c = - 5[/tex]

cancel 49 from both sides:

[tex] \displaystyle c = - 54[/tex]

hence,

  • b=-14
  • C=-54

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Problem:

  • Find the unknown values of b and c in the equation y = -x² + bx + c given its vertex is (-7, -5).

Let's Solve it!

  • This is a problem where you want to able to ake use of tue Vertex Form in order to get the values of b and c fir the standard form of the parabola. Vertex (-7,-5) or (h,k) in the standard form. These are the x and y coordinates of the vertex.

Note the standard form is:

[tex]\quad\quad\quad\quad\tt{ a{x}^{2} + bx + c}[/tex]

Since we have the given of:

[tex]\quad\quad\quad\quad\tt{ a \: = - 1}[/tex]

[tex]\quad\quad\quad\quad\tt{ h = - 7}[/tex]

[tex]\quad\quad\quad\quad\tt{ k \: = - 5}[/tex]

Note the vertex form is:

[tex]\quad\quad\quad\quad\tt{y = a(x - h {)}^{2} } + k[/tex]

[tex]\quad\quad\quad\quad\tt{y = - 1(x - ( - 7) {)}^{2} } + ( - 5)[/tex]

[tex]\quad\quad\quad\quad\tt{y = - 1(x + 7 {)}^{2} } - 5[/tex]

[tex]\quad\quad\quad\quad\tt{ \boxed{y = - {x}^{2} - 14x - 54}}[/tex]

Hence, The answer for b and c is:

[tex]\quad\quad\quad\quad\tt{ \boxed{ \boxed{ \color{magenta}{b = - 14}}}}[/tex]

[tex]\quad\quad\quad\quad\tt{ \boxed{ \boxed{ \color{magenta}{c = - 54}}}}[/tex]

Let's confirm it :

  • You can use x coordinate of the vertex like this,

[tex]\quad\quad\quad\quad\tt{ - 7 = \frac{b}{2(a)} }[/tex]

[tex]\quad\quad\quad\quad\tt{ - 7 = \frac{b}{2(-1)} }[/tex]

[tex]\quad\quad\quad\quad\tt{ - 7 = \frac{b}{-2} }[/tex]

[tex]\quad\quad\quad\quad\tt{ (- 2)( - 7) = - b }[/tex]

[tex]\quad\quad\quad\quad\tt{ \frac{ (- 2)( - 7)}{ - 1} = b }[/tex]

[tex]\quad\quad\quad\quad\tt{ \frac{ 14}{ - 1} = b }[/tex]

[tex]\quad\quad\quad\quad\tt{ \boxed{ - 14 = b }}[/tex]

  • You can also confirm the y coordinate of the vertex in the standard form by plugging the x coordinate.

[tex]\quad\quad\quad\quad\tt{y = - {(-7)}^{2} - 14(-7) - 54}[/tex]

[tex]\quad\quad\quad\quad\tt{y = - 49 +98 - 54}[/tex]

[tex]\quad\quad\quad\quad\tt{y = - 49 +98 - 54}[/tex]

[tex]\quad\quad\quad\quad\tt{y = 49 - 54}[/tex]

[tex]\quad\quad\quad\quad\tt{ \boxed{y = - 5}}[/tex]

So, the final answer for b and c is:

[tex]\quad\quad\quad\quad\tt\huge{ \boxed{ \boxed{ \color{magenta}{b = - 14}}}}[/tex]

[tex]\quad\quad\quad\quad\tt\huge{ \boxed{ \boxed{ \color{magenta}{c = - 54}}}}[/tex]

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