Question

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.f(x)=(x+1)2−1g(x)=(x+1)2+1f(x) = $$(x+1)^2 - 1$$ \quad g(x) = $$(x+1)^2 + 1h(x)=(x$$−1)2+1j(x)=(x−1)2−1h(x) = $$(x-1)^2 + 1$$ \quad j(x) = $$(x-1)^2 - 1$$

Accepted Answer

Identify the vertex of the parabola from the graph. Here, the vertex is given as $$\left(\frac{1}{2}, 2\right). $$Recall the vertex form of a quadratic function: $$f(x) = a(x - h)^2$ + k$$, where $$(h, k) is $$the vertex. Substitute $$h = \frac{1}{2}$$ and $$k = 2$$ into the equation to get $$f(x) = a\left(x - \frac{1}{2}\$$right)^2$$ + 2. $$Use the y-intercept point $$(0, 3) to $$find the value of $$a. $$Substitute $$x = 0$$ and $$f(x) = 3$$ into the vertex form equation: $$3 = a\left(0 - \frac{1}{2}\$$right)^2$$ + 2. $$Simplify the equation to solve for $$a$$: $$3 = a\left(\frac{1}{4}\right) + 2. $$Then isolate $$a by $$subtracting 2 from both sides and dividing by $$\frac{1}{4}. $$Once $$a is $$found, write the final quadratic function equation by substituting $$a$$ back into the vertex form: $$f(x) = a\left(x - \frac{1}{2}\$$right)^2$$ + 2.$$

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.
$f (x) = (x + 1)^{2} - 1$
$h (x) = (x - 1)^{2} + 1$

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Key Concepts

Vertex Form of a Quadratic Function

Using Points to Find the Quadratic Equation

Graph Interpretation and Coordinate Identification

The graph of a quadratic function is given. Write the function's equation, selecting from the following options.f(x)=(x+1)2−1g(x)=(x+1)2+1f(x) = (x+1)^2 - 1 \(\quad\) g(x) = (x+1)^2 + 1h(x)=(x−1)2+1j(x)=(x−1)2−1h(x) = (x-1)^2 + 1 \(\quad\) j(x) = (x-1)^2 - 1