Question

find (a) the axis of symmetry and (b) the vertex of the graph of the function (see example 1). 10. $f(x) = -6x^2 + 24x - 20$ 32. estimate the $y$-intercept of the graph and the maximum or minimum value of the function represented by the graph. $f(x) = -2x^2 + 4x + 3$ (with a graph of the parabola)

Explanation:

For Problem 10: $f(x) = -6x^2 + 24x - 20$

Step1: Find axis of symmetry

For quadratic $ax^2+bx+c$, axis is $x=-\frac{b}{2a}$. Here $a=-6$, $b=24$.
$x = -\frac{24}{2(-6)} = \frac{-24}{-12} = 2$

Step2: Find vertex x-coordinate

Vertex lies on axis of symmetry, so $x=2$.

Step3: Find vertex y-coordinate

Substitute $x=2$ into $f(x)$.
$f(2) = -6(2)^2 + 24(2) - 20 = -6(4) + 48 - 20 = -24 + 48 - 20 = 4$

Step1: Find y-intercept

Y-intercept occurs at $x=0$, substitute $x=0$ into $f(x)$.
$f(0) = -2(0)^2 + 4(0) + 3 = 3$

Step2: Identify max/min value

Quadratic opens downward ($a=-2<0$), so it has a maximum. Use vertex formula for $x$-coordinate: $x=-\frac{b}{2a}=-\frac{4}{2(-2)}=1$.

Step3: Calculate maximum value

Substitute $x=1$ into $f(x)$.
$f(1) = -2(1)^2 + 4(1) + 3 = -2 + 4 + 3 = 5$

Answer:

(a) Axis of symmetry: $x=2$
(b) Vertex: $(2, 4)$

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For Problem 32: $f(x) = -2x^2 + 4x + 3$