The graph below displays a quadratic function. Identify the equation of the functi
and determine the following features of the function: intercepts, vertex, and end
behavior.

Question

The graph below displays a quadratic function. Identify the equation of the functi and determine the following features of the function: intercepts, vertex, and end behavior.

UpStudy AI · Accepted Answer

I'd be happy to help you analyze the quadratic function! However, since I can't see the graph you're referring to, I'll guide you through the general process of identifying the equation of a quadratic function and determining its key features. If you provide specific details from your graph, I can give a more tailored response. ### General Form of a Quadratic Function A quadratic function typically has the form: $$ f(x) = ax^2 + bx + c $$ Alternatively, it can be expressed in **vertex form**: $$ f(x) = a(x - h)^2 + k $$ where: - $$(h, k)$$ is the **vertex** of the parabola. - $$a$$ determines the **opening direction** and **width** of the parabola. ### Steps to Identify the Equation and Features 1. **Determine the Vertex:** - Locate the highest or lowest point on the graph. This point is the vertex $$(h, k)$$. 2. **Find the Y-Intercept:** - The y-intercept occurs where $$x = 0$$. This is the point $$(0, c)$$. 3. **Find the X-Intercepts (Roots):** - These are the points where the graph crosses the x-axis. Solve $$f(x) = 0$$ to find these values. 4. **Determine the Orientation:** - If $$a > 0$$, the parabola opens **upwards**. - If $$a < 0$$, the parabola opens **downwards**. 5. **Find Additional Points (if needed):** - Use other points on the graph to set up equations and solve for unknown coefficients if the vertex form isn't directly applicable. ### Determining the Equation **Using Vertex Form:** If you know the vertex $$(h, k)$$ and another point $$(x, y)$$ on the graph: 1. Start with the vertex form: $$ f(x) = a(x - h)^2 + k $$ 2. Plug in the known point to solve for $$a$$: $$ y = a(x - h)^2 + k $$ 3. Once $$a$$ is known, you have the complete equation. **Conversion to Standard Form:** If needed, expand the vertex form to convert it to the standard form $$ax^2 + bx + c$$. ### Determining Key Features - **Intercepts:** - **Y-Intercept:** Directly read from the graph where $$x = 0$$. - **X-Intercepts:** Solve $$f(x) = 0$$ using the quadratic formula, factoring, or completing the square. - **Vertex:** - Located at $$(h, k)$$ in the vertex form. - **End Behavior:** - If $$a > 0$$: As $$x \to \pm\infty$$, $$f(x) \to +\infty$$. - If $$a < 0$$: As $$x \to \pm\infty$$, $$f(x) \to -\infty$$. ### Example **Given:** - Vertex at $$(2, -3)$$ - Y-intercept at $$(0, 1)$$ - Parabola opens upwards. **Steps:** 1. **Use Vertex Form:** $$ f(x) = a(x - 2)^2 - 3 $$ 2. **Use Y-Intercept to Find $$a$$:** $$ 1 = a(0 - 2)^2 - 3 $$ $$ 1 = 4a - 3 $$ $$ 4a = 4 $$ $$ a = 1 $$ 3. **Equation:** $$ f(x) = (x - 2)^2 - 3 $$ 4. **Convert to Standard Form (optional):** $$ f(x) = x^2 - 4x + 4 - 3 $$ $$ f(x) = x^2 - 4x + 1 $$ 5. **Find X-Intercepts:** $$ x^2 - 4x + 1 = 0 $$ $$ x = \frac{4 \pm \sqrt{16 - 4}}{2} = \frac{4 \pm \sqrt{12}}{2} = 2 \pm \sqrt{3} $$ 6. **End Behavior:** - Since $$a = 1 > 0$$, as $$x \to \pm\infty$$, $$f(x) \to +\infty$$. ### Conclusion To provide a specific equation and accurately determine the features of your quadratic function, please share key details from the graph such as: - Coordinates of the vertex - Y-intercept - X-intercepts (if available) - Any other points that lie on the graph With this information, I can help you find the exact equation and analyze its characteristics comprehensively. To identify the equation of a quadratic function and determine its key features (intercepts, vertex, and end behavior), follow these steps: 1. **Find the Vertex:** - Locate the highest or lowest point on the graph, which is the vertex $$(h, k)$$. 2. **Determine the Y-Intercept:** - Find where the graph crosses the y-axis, which is the point $$(0, c)$$. 3. **Find X-Intercepts:** - Solve $$f(x) = 0$$ to find the points where the graph crosses the x-axis. 4. **Determine the Orientation:** - If the coefficient $$a > 0$$, the parabola opens upwards. - If $$a < 0$$, it opens downwards. 5. **Write the Equation:** - Use the vertex form $$f(x) = a(x - h)^2 + k$$ and solve for $$a$$ using a known point. 6. **Analyze End Behavior:** - As $$x \to \pm\infty$$, $$f(x) \to +\infty$$ if $$a > 0$$, and $$-\infty$$ if $$a < 0$$. **Example:** - **Vertex:** $$(2, -3)$$ - **Y-Intercept:** $$(0, 1)$$ - **Parabola Opens Upwards ($$a > 0$$)** **Equation:** $$ f(x) = (x - 2)^2 - 3 $$ **Features:** - **Intercepts:** Y-intercept at $$(0, 1)$$, X-intercepts at $$x = 2 \pm \sqrt{3}$$ - **Vertex:** $$(2, -3)$$ - **End Behavior:** As $$x \to \pm\infty$$, $$f(x) \to +\infty$$ **Note:** For a precise equation and features, provide specific points from your graph.

The graph below displays a quadratic function. Identify the equation of the functi
and determine the following features of the function: intercepts, vertex, and end
behavior.

Solución de inteligencia artificial de Upstudy

Responder

Solución

Bonus Knowledge

preguntas relacionadas

Latest Algebra Questions