Acceso
Prima
Hogar Banco de preguntas Matemáticas Álgebra
Pregunta
upstudy study bank question image url

Data la funzione \( y=x a x^{2}+b-3 \), determina- re il valore dei coefficienti \( a \) e \( b \) per i quali il grafico della funzione è tangente nel punto di ascissa \( x=1 \) alla retta di equazione \( y=7 x-9 \).

Ask by Johnson Coles. in Italy
Mar 13,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

I = 7/2 e b = -5/2

Solución

1. La funzione è data da \[ y = ax^2 + b - 3 . \] La retta data è \[ y = 7x - 9 . \] 2. Affinché la funzione sia tangente alla retta in \( x=1 \), devono valere due condizioni: - Il punto di tangenza appartiene ad entrambe le curve. - Le pendenze (derivate) sono uguali in \( x=1 \). 3. Calcoliamo il valore di \( y \) per la funzione in \( x=1 \): \[ y(1) = a\cdot(1)^2 + b - 3 = a + b - 3. \] Poiché il punto appartiene anche alla retta, abbiamo: \[ a + b - 3 = 7\cdot1 - 9 = 7 - 9 = -2. \] Quindi: \[ a + b = 1. \quad \text{(Equazione 1)} \] 4. Calcoliamo la derivata della funzione: \[ y'(x) = \frac{d}{dx}(ax^2 + b - 3) = 2ax. \] Alla condizione di tangenza, in \( x=1 \) la derivata deve essere uguale alla pendenza della retta, cioè 7: \[ 2a\cdot1 = 7 \quad \Rightarrow \quad 2a = 7 \quad \Rightarrow \quad a = \frac{7}{2}. \quad \text{(Equazione 2)} \] 5. Sostituendo il valore di \( a \) nell'Equazione 1: \[ \frac{7}{2} + b = 1 \quad \Rightarrow \quad b = 1 - \frac{7}{2} = \frac{2}{2} - \frac{7}{2} = -\frac{5}{2}. \] 6. Quindi, i coefficienti sono: \[ a = \frac{7}{2} \quad \text{e} \quad b = -\frac{5}{2}. \]

Respondido por UpStudy AI y revisado por un tutor profesional

error msg
Explicar
Simplifique esta solución

Mind Expander

Per determinare i coefficienti \( a \) e \( b \), iniziamo a calcolare il valore di \( y \) quando \( x = 1 \) utilizzando la retta \( y = 7x - 9 \): \[ y = 7(1) - 9 = -2. \] Ora sostituiamo \( x = 1 \) nella funzione \( y = ax + x^2 + b - 3 \): \[ -2 = a(1) + (1)^2 + b - 3. \] Da questa equazione otteniamo: \[ -2 = a + 1 + b - 3 \implies a + b - 2 = -2 \implies a + b = 0. \tag{1} \] Passiamo ora a calcolare la derivata della funzione \( y = ax + x^2 + b - 3 \) per trovare la pendenza in \( x = 1 \): \[ \frac{dy}{dx} = a + 2x. \] Sostituendo \( x = 1 \): \[ \frac{dy}{dx} \bigg|_{x=1} = a + 2(1) = a + 2. \] La retta \( y = 7x - 9 \) ha una pendenza costante di \( 7 \). Per la tangente, dobbiamo avere: \[ a + 2 = 7 \implies a = 5. \tag{2} \] Sostituiamo \( a = 5 \) nell'equazione (1): \[ 5 + b = 0 \implies b = -5. \] Quindi, i valori dei coefficienti \( a \) e \( b \) sono \( a = 5 \) e \( b = -5 \).

preguntas relacionadas

Question 6 (1 point) state the total number of complex roots, the possible number of real and imaginary roots and the possible rational roots for each equation. Then find all roots. \[ x^{3}-2 x^{2}-3 x+6=0 \] a \# of complex roots: 6 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1, \pm 2 \pm 3, \pm 6 \) Roots: \( -1, \pm \sqrt{3} \) b \# of complex roots: 6 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1, \pm 2 \pm 3, \pm 6 \) Roots: \( 2, \pm \sqrt{3} \) c \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1, \pm 2 \pm 3, \pm 6 \) Roots \( 2, \pm \sqrt{3} \) d \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1, \pm 2 \pm 3, \pm 6 \) Roots: \( -1, \pm \sqrt{3} \)
Álgebra United States Mar 17, 2025
Question 7 (1 point) State the total number of complex roots, the possible number of real and imaginary roots and the possible rational roots for each equation. Then finc all roots \[ x^{6}-x^{4}-x^{2}+1=0 \] a \# of complex roots: 6 Possible \# of real roots: \( 6,4,2 \), or 0 Possible \# of imaginary roots: \( 6,4,2 \), or 0 Possible rational roots: \( \pm 1 \) Roots: \( \{-1, i,-i\} \) b \# of complex roots: 6 Possible \# of real roots: \( 6,4,2 \), or 0 Possible \# of imaginary roots: \( 6,4,2 \), or 0 Possible rational roots: \( \pm 1 \) Roots: \( \{1 \) mult. \( 2,-1 \) mult. \( 2, i,-i\} \) c \# of complex roots: 6 Possible \# of real roots: \( 6,4,2 \), or 0 Possible \# of imaginary roots: \( 6,4,2 \), or 0 Possible rational roots: \( \pm 1 \) Roots: \( \{1 \) mult. 2, -1 mult. 2\} d \# of complex roots: 6 Possible \# of real roots: \( 6,4,2 \), or 0 Possible \# of imaginary roots: \( 6,4,2 \), or 0 Possible rational roots: \( \pm 1 \) Roots: \( \{1,-1,-2,2, i,-i\} \) Review Answers
Álgebra United States Mar 17, 2025
State the total number of complex roots, the possible number of real and imaginary roots and the possible rational roots for each equation. Then find all roots. \[ x^{3}-1=0 \] a \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) \[ \text { Roots: } \pm 1, \frac{-1+i \sqrt{3}}{2} \] b \[ 1, \frac{-1+i \sqrt{3}}{2}, \frac{-1-i \sqrt{3}}{2} \text { \# of complex roots: } 3 \] Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) Roots: c \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) \[ \text { Roots: }-1, \frac{-1+i \sqrt{2}}{5}, \frac{-1-i \sqrt{2}}{5} \] d \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) Roots: \[ -1, \frac{i \sqrt{3}}{2}, \frac{-i \sqrt{3}}{2} \]
Álgebra United States Mar 17, 2025
himpunan penyelesaian dari sistem persamaan \( 2 x+2 y=4 \) dan \( 3 x+y=6 \) adalah
Álgebra Indonesia Mar 17, 2025
Question 3 (1 point) Using technology and the remainder theorem, find the actual rational zeros of the fun \[ f(x)=x^{3}-14 x^{2}+59 x-70 \] a \( -2,-5,-7 \) b \( 2,5,7 \) c \( \pm 1, \pm 14, \pm 59, \pm 70 \) d \( 1,-14,59,-70 \)
Álgebra United States Mar 17, 2025

Latest Algebra Questions

3. Write out Cayley tables for groups formed by the symmetries of a rectangle and for \( \left(\mathbb{Z}_{4 y},+\right) \). How many elements are in each group? Are the groups the same? Why or why not?
Álgebra United States Mar 17, 2025
himpunan penyelesaian dari sistem persamaan \( 2 x+2 y=4 \) dan \( 3 x+y=6 \) adalah
Álgebra Indonesia Mar 17, 2025
Question 4 (1 point) State the total number of complex roots, the possible number of real and imaginary roots and the possible rational roots for each equation. Then find all roots. \( \begin{array}{l}x^{3}+3 x^{2}-x-3=0 \\ \text { a of complex roots: } 3 \\ \text { Possible \# of real roots: } 3 \text { or } 1 \\ \text { Possible \# of imaginary roots: } 2 \text { or } 0 \\ \text { Possible rational roots: } \pm 1, \pm 3 \\ \text { Roots: }\{-3,1,-1\} \\ \text { \# of complex roots: } 3 \\ \text { Possible \# of real roots: } 3 \text { or } 1 \\ \text { Possible \# of imaginary roots: } 2 \text { or } 0 \\ \text { Possible rational roots: } \pm 1, \pm 3 \\ \text { Roots: }\{3,1,-1\} \\ \text { \# of complex roots: } 3 \\ \text { Possible \# of real roots: } 3 \text { or } 1 \\ \text { Possible \# of imaginary roots: } 2 \text { or } 0 \\ \text { Possible rational roots: } \pm 1, \pm 3 \\ \text { Roots: }\{-3,-1,-1\} \\ \text { \# of complex roots: } 3 \\ \text { Possible \# of real roots: } 3 \text { or } 1 \\ \text { Possible \# of imaginary roots: } 2 \text { or } 0 \\ \text { Possible rational roots: } \pm 1, \pm 3 \\ \text { Roots: }\{3, \text { double root at } 1\}\end{array} \) \( \begin{array}{l}\text { d }\end{array} \)
Álgebra United States Mar 17, 2025
Summarize all pertinent information obtained by applying the graphing strategy and sketch the graph of \( y=f(x) \). \[ f(x)=\frac{x^{2}+13 x+36}{x^{2}+8 x+16} \] tina the aomain or \( \mathrm{r}(\mathrm{X}) \). seiect the correct cnoice deiow ana, it necessary, till in the answer dox to complete your cnoice. A. The domain is all real \( x \), except \( x= \) \( \square \) (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. The domain is all real \( x \). Find the \( x \)-intercepts of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The \( x \)-intercept(s) is/are at \( x= \) \( \square \) . (Type an integer or a simplified fraction. Use a comma to separate answers as needed.) B. There are no \( x \)-intercepts. Find the \( y \)-intercepts of \( f(x) \). Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Álgebra United States Mar 17, 2025
State the total number of complex roots, the possible number of real and imaginary roots and the possible rational roots for each equation. Then find all roots. \[ x^{3}-1=0 \] a \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) \[ \text { Roots: } \pm 1, \frac{-1+i \sqrt{3}}{2} \] b \[ 1, \frac{-1+i \sqrt{3}}{2}, \frac{-1-i \sqrt{3}}{2} \text { \# of complex roots: } 3 \] Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) Roots: c \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) \[ \text { Roots: }-1, \frac{-1+i \sqrt{2}}{5}, \frac{-1-i \sqrt{2}}{5} \] d \# of complex roots: 3 Possible \# of real roots: 3 or 1 Possible \# of imaginary roots: 2 or 0 Possible rational roots: \( \pm 1 \) Roots: \[ -1, \frac{i \sqrt{3}}{2}, \frac{-i \sqrt{3}}{2} \]
Álgebra United States Mar 17, 2025
Anterior Próximo
¡Prueba Premium ahora!
¡Prueba Premium y hazle a Thoth AI preguntas de matemáticas ilimitadas ahora!
Quizas mas tarde Hazte Premium
Estudiar puede ser una verdadera lucha
¿Por qué no estudiarlo en UpStudy?
Seleccione su plan a continuación
Prima

Puedes disfrutar

Empieza ahora
  • Explicaciones paso a paso
  • Tutores expertos en vivo 24/7
  • Número ilimitado de preguntas
  • Sin interrupciones
  • Acceso completo a Respuesta y Solución
  • Acceso completo al chat de PDF, al chat de UpStudy y al chat de navegación
Básico

Totalmente gratis pero limitado

  • Solución limitada
Bienvenido a ¡Estudia ahora!
Inicie sesión para continuar con el recorrido de Thoth AI Chat
Continuar con correo electrónico
O continuar con
Al hacer clic en "Iniciar sesión", acepta nuestros términos y condiciones. Términos de Uso & Política de privacidad