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Which graph best represents
y=2x^(2)-5x-3 ?

Which graph best represents $y=2x^{2}-5x-3$ ?

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Find the vertex of the transformed function

Full solution

Q. Which graph best represents

y=2x^{2}-5x-3

?

Identify Function Type: Identify the type of function and its general shape. $\newline$ We're dealing with a quadratic function, which is represented by a parabola. The coefficient of $x^2$ is $2$ , which is positive, so the parabola opens upwards.
Find Vertex: Find the vertex of the parabola using the vertex formula $h = -\frac{b}{2a}$ and $k = f(h)$ . $\newline$ For the function y = $2$ x^ $2$ - $5$ x - $3$ , a = $2$ and b = $-5$ . $\newline$ Calculate h: $h = -\frac{-5}{2*2} = \frac{5}{4}$ . $\newline$ Then, substitute h back into the function to find k: $\newline$ $k = 2(\frac{5}{4})^2 - 5(\frac{5}{4}) - 3 = 2(\frac{25}{16}) - \frac{25}{4} - 3 = \frac{50}{16} - \frac{100}{16} - \frac{48}{16} = -\frac{98}{16} = -6.125$ .
Determine Y-Intercept: Determine the y-intercept of the function. $\newline$ The y-intercept occurs where $x = 0$ . Substituting $x = 0$ in $y = 2x^2 - 5x - 3$ gives: $\newline$ $y = 2(0)^2 - 5(0) - 3 = -3$ .
Check Symmetry & Points: Check for symmetry and additional points for a more accurate graph. $\newline$ The axis of symmetry is $x = 1.25$ (from $h$ value). Calculate $y$ values for points around the vertex, like $x = 0, 1, 2, 3$ . $\newline$ $y(0) = -3$ (already calculated), $\newline$ $y(1) = 2(1)^2 - 5(1) - 3 = 2 - 5 - 3 = -6$ , $\newline$ $y(2) = 2(2)^2 - 5(2) - 3 = 8 - 10 - 3 = -5$ , $\newline$ $y(3) = 2(3)^2 - 5(3) - 3 = 18 - 15 - 3 = 0$ .

More problems from Find the vertex of the transformed function

Question

h

is defined over the real numbers. This table gives a few values of

h

\newline

\begin{tabular}{lllll}

\newline

x

-6

1

-6

01

-6

001

-5

9

\newline

h(x)

-0

25

-0

74

-0

98

-1

0

\newline

\newline

What is a reasonable estimate for

\lim _{x \rightarrow-6} h(x)

\newline

1

\newline

-6

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-2

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-1

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