Question 262·Hard·Math

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The graph of $y=ax^2+bx+c$ is shown, where $a$ , $b$ , and $c$ are constants. Points $A$ , $B$ , and $C$ are labeled on the graph.

Which choice gives the value of $b$ ?

For a quadratic shown with labeled points, equal-height points reveal a mirror line through their horizontal midpoint. In $ax^2+bx+c$ , that line is $x=-\frac{b}{2a}$ , which relates the requested coefficient to $a$ . A point on the $y$ -axis gives $c$ ; one more point leaves an equation Desmos can solve. The midpoint is the axis, not the coefficient $b$ .

Hints

Hint 1
A parabola mirrors across a vertical axis of symmetry. Points at the same height sit equally far from that line. What is halfway between the $x$ -coordinates of $A$ and $B$ ?
Hint 2
For $y=ax^2+bx+c$ , the axis has $x=-\frac{b}{2a}$ . Set this equal to the midpoint you found, then write $b$ in terms of $a$ .
Hint 3
At $x=0$ , both terms containing $x$ disappear, so point $C$ gives the constant term $c$ . Use point $A$ to make an equation for the one unknown left, $a$ .

Step-by-step

Use symmetry, then solve for the coefficient
Step 1
Find the mirror line
Points $A=(-1,6)$ and $B=(5,6)$ have the same height. A parabola mirrors across its axis of symmetry, so that vertical line is halfway between their $x$ -coordinates. Find the midpoint:
$x=\frac{-1+5}{2}=2.$
Step 2
Relate $b$ to $a$
For $y=ax^2+bx+c$ , the axis is $x=-\frac{b}{2a}$ . Its location is $2$ , so set them equal: $2=-\frac{b}{2a}$ . Multiply by $2a$ : $4a=-b$ . Multiply by $-1$ : $b=-4a$ . The axis is $2$ , but $b$ is not $2$ ; it also depends on $a$ .
Step 3
Get $c$ from the point on the $y$ -axis
Point $C=(0,2)$ has input $x=0$ . Put that input and output into the given equation: $2=a(0)^2+b(0)+c$ . Both terms containing $x$ vanish:
$c=2.$
Step 4
Turn point $A$ into an equation
Point $A=(-1,6)$ means the output is $6$ when the input is $-1$ . Substitute both coordinates: $6=a(-1)^2+b(-1)+c$ . Simplify the square and the negative input:
$6=a-b+c.$
Step 5
Leave only $a$ unknown
Replace $b$ with $-4a$ and $c$ with $2$ in the equation from $A$ : $6=a-(-4a)+2$ . Remove the double negative: $6=a+4a+2$ . Combine the terms with $a$ :
$6=5a+2.$
Step 6
Solve the one-unknown equation
Type $5x_1+2\sim6$ , letting $x_1$ stand for $a$ . The regression symbol $\sim$ tells Desmos to find the unknown rather than make a slider. Under PARAMETERS, Desmos shows $x_1=0.8$ , so $a=0.8$ .
Step 7
Calculate the requested coefficient
Since $b=-4a$ , type $-4x_1$ on a new Desmos line. It prints $-3.2$ ; use that line’s fraction button to see $-\frac{16}{5}$ . That is the coefficient of $x$ , $b$ . Choice A.

Question 262·Hard·Math

Strategy

Hints

Step-by-step

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Strategy

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Step-by-step

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