Question

For the function, the vertex of the function's graph is given. Find the unknown coefficients.

y=x^2+bx+c ; (-2,4)

Answer 1

$\bf \textit{vertex of a vertical parabola, using coefficients} \\\\ y=\stackrel{\stackrel{a}{\downarrow }}{1}x^2\stackrel{\stackrel{b}{\downarrow }}{+b}x\stackrel{\stackrel{c}{\downarrow }}{+c} \qquad \qquad \left(-\cfrac{ b}{2 a}~~~~ ,~~~~ c-\cfrac{ b^2}{4 a}\right)~~=~~(-2,4) \\\\[-0.35em] ~\dotfill\\\\ -\cfrac{b}{2a}=-2\implies -\cfrac{b}{2(1)}=-2\implies -b=-4\implies \boxed{b=4} \\\\[-0.35em] ~\dotfill$

$\bf c-\cfrac{ b^2}{4 a}=4\implies c-\cfrac{\boxed{4}^2}{4(1)}=4\implies c-\cfrac{16}{4}=4 \\\\\\ c-4=4\implies \boxed{c=8} \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ ~\hfill y=x^2+4x+8~\hfill$

Answer 2

The answer is y = x2 + 4x + 8

The graph of a quadratic equation (y = ax2 + bx + c) is the shape of a parabola. A parabola looks like a U or an upside-down U. The vertex of a quadratic equation is the maximum or minimum point on the equation's parabola.

Which function's graph has a vertex?

the quadratic function

One important feature of the parabola is that it has an extreme point, called the vertex. If the parabola opens up, the vertex represents the lowest point on the graph or the minimum value of the quadratic function. If the parabola opens down, the vertex represents the highest point on the graph or the maximum value.

How do you find the vertex of a graph?

Finding Vertex of a Parabola From Standard Form

Step - 1: Compare the equation of the parabola with the standard form y = ax2 + bx + c. ...

Step - 2: Find the x-coordinate of the vertex using the formula, h = -b/2a. ...

Step - 3: To find the y-coordinate (k) of the vertex, substitute x = h in the expression ax2+ bx + c.How do you find the vertex of a graph?

Finding Vertex of a Parabola From Standard Form

Step - 1: Compare the equation of the parabola with the standard form y = ax2 + bx + c. ...

Step - 2: Find the x-coordinate of the vertex using the formula, h = -b/2a. ...

Step - 3: To find the y-coordinate (k) of the vertex, substitute x = h in the expression ax2+ bx + c

Learn more about  the vertex of the function's graph

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