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AI
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    - Fracciones
    - Decimales
    - Exponentes
    - Radicales
    - Proporciones
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  - Álgebra
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    - Limits
    - Serie
  - Trigonometría
  - Geometría
  - Estadística y Probabilidad
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- Calculadoras
  - Geometría
  - Estadística y probabilidad
    - Desviación estándar
    - Media y mediana y moda y rango
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    - Razón
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  - Trigonometría
    - Funciones trigonométricas
    - Lista de identidades trigonométricas
  - Aritmética
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Acceso

**Step 1. Write the Equation and Identify Coefficients:**  
The quadratic function is given by  
\[
y = x^2 - 1.
\]  
This is in the standard form  
\[
y = ax^2 + bx + c,
\]  
where  
\[
a = 1,\quad b = 0,\quad c = -1.
\]

**Step 2. Determine the Vertex:**  
The vertex of a parabola in standard form has \(x\)-coordinate  
\[
x = -\frac{b}{2a}.
\]  
Substitute the coefficients:  
\[
x = -\frac{0}{2 \times 1} = 0.
\]  
Then, substitute \(x = 0\) into the equation to find the \(y\)-coordinate:  
\[
y = (0)^2 - 1 = -1.
\]  
Thus, the vertex is at  
\[
(0, -1).
\]

**Step 3. Identify the Axis of Symmetry:**  
The axis of symmetry is the vertical line passing through the vertex. Its equation is  
\[
x = 0.
\]

**Step 4. Find the \(x\)-Intercepts:**  
Set \(y = 0\) and solve for \(x\):  
\[
0 = x^2 - 1.
\]  
Rearrange the equation:  
\[
x^2 = 1.
\]  
Taking the square root of both sides yields:  
\[
x = \pm 1.
\]  
Thus, the \(x\)-intercepts are  
\[
(1, 0) \quad \text{and} \quad (-1, 0).
\]

**Step 5. Find the \(y\)-Intercept:**  
Set \(x = 0\) in the equation:  
\[
y = (0)^2 - 1 = -1.
\]  
So the \(y\)-intercept is at  
\[
(0, -1).
\]

**Step 6. Describe the Graph:**  
- The parabola opens upward since \(a = 1 > 0\).  
- The vertex is the lowest point, at \((0, -1)\).  
- The graph is symmetric about the line \(x = 0\).  
- It crosses the \(x\)-axis at \((1, 0)\) and \((-1, 0)\), and the \(y\)-axis at \((0, -1)\).

Hogar Banco de preguntas Matemáticas Álgebra

Pregunta

c.

Ask by Higgins Schneider. in the United States

Mar 25,2025

Solución de inteligencia artificial de Upstudy

Respuesta verificada por el tutor

Responder

The graph of

is a parabola opening upward with its vertex at

. It crosses the

-axis at

and

, and the

-axis at

Solución

Step 1. Write the Equation and Identify Coefficients:
The quadratic function is given by

This is in the standard form

where

Step 2. Determine the Vertex:
The vertex of a parabola in standard form has

-coordinate

Substitute the coefficients:

Then, substitute

into the equation to find the

-coordinate:

Thus, the vertex is at

Step 3. Identify the Axis of Symmetry:
The axis of symmetry is the vertical line passing through the vertex. Its equation is

Step 4. Find the -Intercepts:
Set

and solve for

Rearrange the equation:

Taking the square root of both sides yields:

Thus, the

-intercepts are

Step 5. Find the -Intercept:
Set

in the equation:

So the

-intercept is at

Step 6. Describe the Graph:

The parabola opens upward since .
The vertex is the lowest point, at .
The graph is symmetric about the line .
It crosses the -axis at and , and the -axis at .

Respondido por UpStudy AI y revisado por un tutor profesional

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Simplifique esta solución

Beyond the Answer

Did you know that the equation

represents a parabola? This particular parabola opens upwards, and its vertex is located at the point (0, -1). Parabolas like this one are not just abstract concepts; they can be seen in real-life structures such as bridges and satellite dishes, where the shape helps in distributing weight and focusing signals efficiently!

When working with this equation, a common mistake is forgetting to plot the vertex and the axis of symmetry. The vertex is crucial because it indicates the minimum point for curves that open upwards. Always remember to check the y-intercept (when

) and other key points to sketch the graph accurately!

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