Write equation in standard 19) $$ \begin{array}{cll}x & y & ext { form. } \ -2 & 16 \ -1 & 7 \ 0 & 4 & \ 1 & 7 & \ 2 & 16\end{array} $$

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Write equation in standard 19) $$ \begin{array}{cll}x & y & 	ext { form. } \ -2 & 16 \ -1 & 7 \ 0 & 4 & \ 1 & 7 & \ 2 & 16\end{array} $$

UpStudy AI · Accepted Answer

We are given the points  
$$
(-2,16),\ (-1,7),\ (0,4),\ (1,7),\ (2,16)
$$  
and we wish to find the quadratic equation in standard form,  
$$
y = ax^2 + bx + c.
$$

**Step 1. Determine symmetry.**  
Notice that the points $$(-2,16)$$ and $$(2,16)$$ as well as $$(-1,7)$$ and $$(1,7)$$ have the same $$y$$-values. This symmetry about the $$y$$-axis suggests the quadratic is even, so the coefficient $$b=0$$. Thus, the equation simplifies to  
$$
y = ax^2 + c.
$$

**Step 2. Find $$c$$.**  
Using the point where $$x=0$$, we have  
$$
4 = a\cdot0^2 + c \quad \Longrightarrow \quad c = 4.
$$

**Step 3. Find $$a$$.**  
Using any other point, for example $$(1,7)$$, substitute $$x=1$$ and $$y=7$$ into the equation:  
$$
7 = a\cdot1^2 + 4 \quad \Longrightarrow \quad a = 7 - 4 = 3.
$$

**Step 4. Write the equation.**  
Substitute $$a=3$$ and $$c=4$$ into the simplified form:  
$$
y = 3x^2 + 4.
$$

Thus, the equation in standard form is  
$$
y = 3x^2 + 4.
$$
The equation in standard form is $$ y = 3x^2 + 4 $$.

Write equation in standard
19)

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