When the quadratic function is graphed in the -plane, where , its vertex is . One of the -intercepts of this graph is . What is the other -intercept of the graph?

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SAT Nonlinear Functions Question 72 | Free SAT Prep | Aniko

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Question 72·Hard·Nonlinear Functions

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When the quadratic function $g$ is graphed in the $xy$ -plane, where $y = g(x)$ , its vertex is $\left(\dfrac{5}{2},\,-9\right)$ . One of the $x$ -intercepts of this graph is $\left(\dfrac{11}{4},0\right)$ . What is the other $x$ -intercept of the graph?

When a quadratic question gives you the vertex and one $x$ -intercept and asks for the other $x$ -intercept, avoid writing the full equation or using the quadratic formula. Instead, use the fact that the vertex lies on the axis of symmetry: its $x$ -coordinate is the average of the $x$ -coordinates of the intercepts. Either (1) compute the horizontal distance between the vertex and the known intercept and reflect it to the other side, or (2) set up $(x_1 + x_2)/2 = h$ with $h$ as the vertex’s $x$ -value and solve for the missing $x$ quickly using basic fraction arithmetic.

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Think about the axis of symmetry

For a parabola given by a quadratic function, what vertical line passes through the vertex and splits the graph into two mirror-image halves?

Relate the intercepts to the vertex

The two $x$ -intercepts lie on opposite sides of the axis of symmetry, and the vertex is exactly halfway between them. How can you express "halfway between" using an average of the two $x$ -coordinates?

Use equal distances from the axis

Find the horizontal distance between the vertex’s $x$ -coordinate and the known intercept’s $x$ -coordinate. Then place the unknown intercept the same distance on the other side of the axis of symmetry.

Desmos Guide

Compute the other x-intercept using the midpoint idea

Because the vertex’s $x$ -coordinate is the average of the two intercepts, the other $x$ -coordinate is $2\cdot(5/2) - 11/4$ . In a new expression line in Desmos, type 2*(5/2) - 11/4. The value Desmos gives is the $x$ -coordinate of the other $x$ -intercept; combine this value with $0$ to form the ordered pair.

Step-by-step Explanation

Use the vertex to find the axis of symmetry

For a quadratic $y=g(x)$ , the vertex has $x$ -coordinate equal to the axis of symmetry. Since the vertex is at $(5/2,-9)$ , the axis of symmetry is the vertical line

x = \frac{5}{2}.

This line is exactly halfway between the two $x$ -intercepts.

Use symmetry of the parabola

On a parabola, points that have the same $y$ -value and lie on opposite sides of the axis of symmetry are the same horizontal distance from that axis. The two $x$ -intercepts both have $y=0$ , so their $x$ -coordinates are equally spaced on either side of $x=5/2$ .

Equivalently, if $x_1$ and $x_2$ are the $x$ -intercepts, then their average is the $x$ -coordinate of the vertex:

\frac{x_1 + x_2}{2} = \frac{5}{2}.

Find the horizontal distance from the vertex to the known intercept

We are told that one $x$ -intercept is at $(11/4,0)$ . Find how far this is from the axis of symmetry $x=5/2$ :

\text{distance} = \frac{11}{4} - \frac{5}{2}.

Write $5/2$ with denominator $4$ :

\frac{5}{2} = \frac{10}{4},

\text{distance} = \frac{11}{4} - \frac{10}{4} = \frac{1}{4}.

So the known intercept is $1/4$ unit to the right of the axis of symmetry.

Place the other intercept the same distance on the other side

The other $x$ -intercept must be $1/4$ unit to the left of $x=5/2$ . Subtract this distance:

\frac{5}{2} - \frac{1}{4}.

Again write $5/2$ as $10/4$ :

\frac{5}{2} = \frac{10}{4}, \quad \frac{10}{4} - \frac{1}{4} = \frac{9}{4}.

Thus the other $x$ -intercept has $x$ -coordinate $9/4$ , and its coordinates are

(9/4,0).

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation